UC-NRLF 


^B    SE&    33b 


Plane  and  Solid  Geometry 


INDUCTIVE   METHOD 


<6  (o 


IN  MEMORIAM 
FLORIAN  CAJORI 


PLANE  AN-PySi&tfp  •••J-' 

GEOMETRV 


INDUCTIVE  METHOD 


A.  A.  DODD,  M.S.D..  S.B.,  and  B.  T.  CHACE, 

Teachers  of  Mathematics  in  the  Manual  Training  High  School, 
Kansas  City,  Mo. 


PRESS  OF 

HUDSON-KIMBERIyY  PUBLISHING  CO., 

KANSAS  CITY,  MO. 


h^ 


Copyrighted  by 

HUDSON-KIMBERLY  PUBLISHING  CO. 

1898. 


CAJORl 


PREFACE. 


The  purpose  of  this  elementary  treatment  of  Plane  Geom- 
eiry  is  to  give  to  the  pupils  of  the  Kansas  City  Manual  Train- 
ing High  School  a  course  in  geometric  reasoning  based  prin- 
cipally on  the  Inductive  method  of  presentation. 

In  the  beginning,  by  many  questions  and  suggestions 
carefully  arranged,  the  pupil  is  led  to  grasp  some  of  the  fun- 
damental ideas  of  Geometry,  and  in  the  same  manner  the  first 
propositions  are  established. 

Instead  of  giving  the  formal  proposition  at  the  begin- 
ning of  a  demonstration  and  then  the  proof,  the  pupil  reaches 
the  general  truth  as  a  result.  This  result  or  proposition  is  to 
be  neatly  written  and  numbered  on  the  blank  pages  at  the 
back  of  the  book.  With  the  lessons  prepared,  partly  by  an- 
swering the  questions,  and  under  the  direction  of  skillful  teach- 
ers, it  is  hoped  that  the  majority  of  the  pupils  will  be  able  to 
make  logical  deductions  from  data  given  and  to  become  self- 
reliant.  It  is  not  the  amount  of  knowledge  possessed,  but 
rather  the  method  by  which  the  knowledge  is  gained,  which  is 
the  important  thing.  What  power  results  from  the  investiga- 
tion of  the  truths  in  Geometry  is  the  all-important  question. 

No  boy  or  girl  will  learn  to  ride  a  bicycle  by  memorizing 
the  most  carefully  prepared  directions.  Actual  struggle  and 
persevering  efiforts  with  thoughtful  direction  bring  skill  to 
the  learner. 

No  amount  of  memorizing  of  r&2iSomng processes  will  make 
a  pupil  proficient  in  reasoning.  It  will  tend  to  keep  him  from 
using  his  originative  and  reasoning  powers. 

It  is  supposed  that  the  pupils  who  take  this  course  have 
had,  at  least,  one  term  in  Inventional  or  Constructional  Geom- 
etry; if  the  class  has  not  had  this  preliminary  work,  the 
teacher  should  spend  some  time  in  introducing  the  subject 

918283 


concretely,  familiarizing  the  pupils  with  dividers  and  rule; 
to  pupils  thus  prepared  the  construction  of  most  plane  fig- 
ures and  the  grasping  of  simple  geometric  ideas  should  pre- 
sent no  serious  difficulties.  The  energies  of  the  pupil  can  be 
directed  to  the  processes  of  reasoning.  In  this  school  espe- 
cially do  we  try  to  teach  by  doing,  guided  by  skilled  directors. 
What  the  actual  experimental  work  in  the  laboratory,  shop,  or 
cooking-room  is  to  the  theory  of  the  science  taught,  so  is  the 
original  solution  or  demonstration  to  the  principles  and  theo- 
rems in  Geometry. 

For  this  reason,  numerous  graded  exercises  are  given  along 
with  the  propositions.  Many  of  these  exercises  are  intended  to 
make  the  pupil  feel  that  Geometry  is  a  science  which  has  to 
do  with  common  afifairs.  An  early  introduction  of  the  prop- 
erties of  the  triangle  is  easily  made  to  the  pupil  by  the  method 
of  superposition;  and  the  utility  of  this  method  is  of  great  value 
in  acquiring  other  geometric  truths. 

Accuracy,  neatness  in  demonstration,  and  the  giving  of 
authority  are  insisted  on  in  all  the  work  to  be  done.  Inde- 
pendent solutions  are  encouraged. 

Nearly  every  English  Elementary  Geometry  has  been 
made  use  of  in  securing  suggestions  and  hints  on  demonstra- 
tions, but  the  method  of  reaching  the  general  truths  of 
Elementary  Geometry  is  unique  and  is  believed  to  be  on 
the  laboratory  method  of  teaching.  It  must  not  be  forgotten 
that  those  who  take  this  course  should  have  finished  a  book 
like  Spencer's  Inventional  Geometry,  or  Nichols'  Construc- 
tional Geometry,  or  Hornbrook's  Concrete  Geometry,  or  to 
have  had  a  good  preliminary  introduction  to  Demonstrative 
Geometry.  In  most  instances  the  question  whose  answer  is 
the  proposition  required  is  printed  in  pica,  thus  helping  the 
pupil  to  keep  the  main  question  in  mind. 

A.  A.  DoDD. 

B.  T.  Chack. 
Manual  Training  High  School,  Kansas  City,  Mo. 


SUGGESTIONS  TO  TEACHERS. 


It  is  of  the  greatest  importance  that  the  pupil  clearly 
understand  the  status  of  his  work  in  Constructional  Geometry. 
Of  the  three  books  mentioned  in  the  Preface,  Nichols'  is  the 
only  one  which  attempts  pure  demonstrations.  But  even 
these  cannot  be  accepted  (see  pages  32,  41,  48,  54,  etc.), 
since  the  problems,  (1)  To  construct  an  angle  equal  to  a  given 
angle;  (2)  .To  bisect  a  given  angle,  etc.,  have  not  yet  been 
proved  to  be  geometrically  true.  In  Proposition  I.  he  does  not 
know  that  he  has  two  triangles  in  which  two  respective  sides 
and  included  angles  are  equal  because  he  constructed  them 
equal,  but  because  they  are  so  given  in  the  hypothesis. 

In  beginning  the  tormal  demonstration  of  Proposition  I, 
the  teacher  should  clearly  outline  the  method  of  proof  2lX\^  by 
numerous  exercises  and  illustrations  see  that  this  is  fully 
understood  by  the  pupils.  Let  the  pupils  understand  that  in 
Proposition  I.  and  others  the  questions  asked  about  the  figures 
are  no  part  of  the  demonstration — they  are  asked  simply  to  lead 
the  piipil  to  discover  that  proof  and  the  steps  in  it  which  he 
is  required  to  give  in  logical  order. 


ORDER  OF  PROOF. 


I.  General  Enunciition,  which  should  be  clearly  shown 
to  consist  of  two  parts,  the  hypothesis  (or  supposition),  and  the 
coticlusion.  Thus  in  Proposition  I.  we  have:  Hypothesis. — If 
two  sides  and  the  included  angle  of  any  triangle  are  equal, 
respectively,  to  two  sides  and  the  included  angle  of  another 
triangle,  conclusion.,  the  triangles  are  equal. 

II.  The  Particular  Enunciation,  which  refers  to  the  par- 
ticular figure  or  figures  which  fulfill  all  the  given  conditions 
of  the  General  Enunciation.     Thus  in  Proposition  I.: 

Given:    The  triangles  ABC  and  D  E  F,  in  which  A  B,  A  C, 


and    Z    B  A  C,  respectively;  equal  D  E,  D  F,  and  Z  E  D  F. 
To  prove:     The  triangles  ABC  and  D  E  F  equal. 

III.  The  Consiructiony  which  consivSts  of  the  drawing  of 
aid  lines,  superposition  of  figures,  etc.  Here  the  authority 
for  the  work  should  be  shown  to  rest  upon  the  geometric 
postulates  which  are  discussed  in  the  text.  In  problems  the 
Construction  is  given  a  prominent  place.  At  all  times  the 
work  should  be  done  with  the  greatest  care  and  accuracy. 

IV.  The  Demonstration,  which  is  shown  to  rest  solidly 
upon  definitions,  axioms,  and  previously  proved  theorems 
and  problems  previously  constructed  and  proved. 

It  is  of  vital  importance  that  the  pupil  fully  understand 
that  the  truth  set  to  be  proved  in  the  Particular  Enunciation  is 
not  established  until  the  very  best  authority  has  been  given,  and 
then  the  pupil  should  be  led  to  see  clearly  just  how  the  con- 
clusion of  the  General  Enunciation  follows  the  proof  of  the 
Particular  Enunciation.  The  pupil  must  not  be  permitted  to 
conclude  that  his  work  in  Constructional  Geometry  is  useless 
because  he  can  not  use  it  as  authority  for  his  present  work. 
The  definitions  and  axioms  there  given  are  authority  in  the  pres- 
ent work.  (The  fundamental  notions  there  developed  will  be 
of  great  value  in  the  work  in  hand.)  Truths  which  are  there 
proved  and  are  shown  to  depend  solely  upon  the  definition 
for  authority  or  which  followed  from  the  application  of  axioms 
are  i?i  full  J  or  ce  here.  But  he  must  understand  once  for  all 
that  the  further  truths  which  he  discovered  and  carefully 
tested  with  instruments  must  be  now  formally  established  by 
the  strictest  of  logical  reasoning.  He  has  studied  Practical 
Geometry. 

He  can  be  shown  that  the  fundamental  notions  there 
developed  will  be  of  the  greatest  value  in  the  present  sci- 
ence of  reasoning,  which  ,deals  with  the  truths  there  discov- 
ered and  practically  used. 

When  the  study  of  Concrete  Geometry,  as  recommended 
by  the  Committee  of  Ten,  has  been  widely  introduced  into  the 
grammar  grades  of  our  schools,  this  work  can  be  begun  in 
the  beginning  of  the  High  School. 


TABLE  OF  CONTENTS. 


Page 

Symbols  and  Abbreviations 10 

Preliminary  Questions 11 

Introduction 11 

Postulates 12 

Axioms 13 

Definitions 14 

Axioms  and  Postulates 21 

Suggestions  to  Pupils 22 

PI.ANE  GEOMETRY. 

Book  I.    Lines  and  Rectilinear  Figures,  Problems. 

Angles,  Discussion  of  the  Demonstration 24 

Triangles,  Model  Demonstrations,  Definitions 26 

Problems,  Perpendiculars,  Angles 45 

Perpendiculars,  Right  Triangles 55 

Parallels 59 

Triangles 63 

Parallelograms,  Definitions 67 

Polygons,  Bisectors,  Transversals,  Loci 74 

Book  II.    Circles,  Problems. 

Definitions,  Chords,  Problems 87 

Tangents,  Problems,  Inscribed  Angles,  Secants 95 

Problems 110 


8  GEOMETRY. 

Book  III.     Proportion,  Similar  Figures. 

Measurement,  Ratio 113 

Proportion 118 

Limits 129 

Proportional  I^ines 1 35 

Similar  Figures,  Problems 141 

Exercises,  Problems. 161 

Book  IV.     Areas. 

Quadrilaterals 167 

Relations  of  Homologous  Parts  of  Similar  Figures 179 

Triangles,  Areas 1 85 

Supplementary  Exercises,  Problems 1 89 

Moulding  of  Polj^gons,  Exercises 199 

Book  V.    Measurement  of  the  Circle. 

Symmetry,  Regular  Polygons 202 

Problems 209 

Maxima  and  Minima  of  Plane  Figures 220 

Selected  Examination  Papers 229 

SOI.ID  GEOMETRY. 

Book  VI.     Planes  and  Polyedral  Angles. 

Preliminary  Discussion,  Lines  and  Planes  in  Space 236 

Definitions,  Lines  and  Planes  in  Space 248 

Diedral  Angles 254 

Polyedral  Angles 263 

Book  VII.  Polyedrons. 

Definitions 271 

Prisms 274 

Parallelopipeds ......  279 


TABLE  OF  CONTENTS.  9 

Pyramids 289 

Regular  Polyedrons,  Problems,  Kuler's  Theorem 308 

Book  VIII.    The  Three  Round  Bodies. 

The  Cylinder *. 316 

The  Cone 328 

The  Sphere 340 

Spherical  Angles  and  Polygons 353 

Spherical  Measurements 368 

Exercises , 381 

Formulas  and  Numerical  Tables 386 

Tables  of  Denominate  Numbers 388 

Biographical  Notes ....  392 

Index 399 


10 


GEOMETRY. 


Symbols  and  Abbreviations, 


z 

=  angle. 

/.s 

=  angles. 

A 

=  triangle. 

As 

=  triangles. 

n 

=  rectangle. 

□  s 

=  rectangles. 

o 

=  circle. 

Os 

=  circles. 

- 

=  arc. 

-^s 

=  arcs. 

± 

=  perpendicular. 

±s 

=  perpendiculars. 

II 

=  parallel. 

II  s 

^  parallels. 

a 

=  parallelogram. 

as 

=  parallelograms. 

> 

==  is,  or  are,  greater  than 

< 

=  is,  or  are,  less  than. 

.*. 

=  therefore. 

•/ 

=  since,  or  because. 

-£- 

=  approaches  as  a  limit. 

t 

=  foot  or  feet. 

ft 

=  inch  or  inches. 

Isos 

.  ^  isosceles. 

Ax. 

=  axiom. 

Cor.    =:  corollary. 

Iden.  =  identical. 

Rt.      =  right. 

Ex.     =  exercise. 

Auth.=  give  authority=why? 

Hyp.  =  hypothesis. 

St.       =  straight 

Pt.         rrr  point. 

Pis.     =  points. 

P.   P.  =  previous  proposition 

Prop.  =  proposition. 

Def.    =  definition. 

Sug.    =  suggestion. 

Sup.   =  supplementary. 

Adj.    ^  adjacent. 

Ext.    =  exterior. 

Int.     =  interior. 

Alt.    ==  alternate. 

Ext. -int. =  exterior-interior. 

Alt.-int.  =  alternate-interior. 

Q.  E.  D.  =:  quod  erat  demon- 
strandum— which  was  to 
be  proved. 

Q.  E.  F.  =  quod  erat  facien- 
dum— which  was  to  be 
done. 


PRBLIMINASflT. ;  ;  \  11 


INTRODUCTION. 

What  dimensions  has  a  solid? 

What  are  the  boundaries  of  a  solid?     Give  examples. 
What  are  the  dimensions  of  a  surface?     Give  examples. 
Give  the  boundaries  of  a  surface.     Illustrate. 
What  dimension  has  a  line?      What  are  its  boundaries? 
Can  we  apply  the  word  '^dimension"  to  a  point? 

1 .  Think  of  a  square  lying  horizontally.  Raise  it  verti- 
cally. What  solid  is  describ  id?  What  surfaces  do  the  sides  of 
the  square  describe? 

2.  Think  of  a  circle  lying  horizontally.  Raise  it  verti- 
cally. What  solid  is  described?  What  surface  is  described  by 
the  circumference?  Revolve  a  rectangle  about  one  of  its  sides 
as  an  axis.     What  solid  is  generated?     What  surfaces? 

3.  Imagine  a  semicircle  revolved  about  its  diameter. 
What  solid  is  formed  by  this  revolution?  What  surface  is 
generated  by  the  semi-circumference?  What  does  the  revolu- 
tion of  the  diameter  generate? 

4.  As  you  fill  a  vessel  with  water,  what  is  the  solid 
traced  by  the  surface  of  the  water? 

If  a  point  move  through  space,  what  will  it  describe? 

If  a  line  move  keeping  parallel  to  its  original  position, 
what  will  it  generate? 

If  a  plane  move  at  right  angles  to  its  original  position, 
what  will  it  generate? 

Revolve  a  line  about  one  end  as  a  center.  What  surface 
is  described  by  the  line?    What  is  described  by  the  other  end? 

Revolve  a  right  triangle  about  each  side  in  order.  De- 
scribe each  solid  and  each  surface  formed. 

Revolve  an  obtuse  triangle  about  each  side  in  order.  De- 
scribe the  solids  and  the  surfaces  formed. 

Can  you  imagine  a  hollow  glass  cube?  Can  you  picture 
other  hollow  glass  figures?     Give  examples. 


12  ^^ffi(>METRY. 

Can  you  irflagine  the  cube,  were  the  glass  cube  shattered? 
Can  you  see  at  cube  with  your  eyes  closed?  Do  you  see  the 
upper  surface?  the  lower  surface?  the  upper  front  edge? 
the  lower  front  edge?  the  other  edges?  the  upper  right 
front  corner?     the  other  corners? 

Think  of  a  cube  bisected.  What  kind  of  surfaces  bound 
the  parts? 

In  how  man}^  ways  can  you  think  of  bisecting  a  cylinder? 
How  many  ways  of  bisecting  a  sphere? 

Think  of  other  solids;  conceive  them  bisected.  What  new 
solids  and  plane  figures  are  thus  formed? 

Write  short,  clear  definitions  of  solid,  surface,  line,  and 
point. 

If  you  can  think  of  a  cube  apart  from  the  material,  of  its 
sides,  of  its  edges,  and  its  corners,  you  have  a  geometrical  con- 
cept of  a  cube.  If  you  can  think  of  a  surface  apart  from  the 
solid  which  it  bounds,  you  have  a  geometrical  co7icept  of  a  sur- 
face. If  you  can  think  of  a  line  apart  from  the  surface  which 
it  bounds,  you  have  a  geometrical  concept  of  a  line. 

If  you  can  think  of  a  point  apart  from  the  extremities  of 
a  line  or  the  intersection  of  two  or  more  lines,  you  have  a 
notion  of  the  geometrical  point. 

Can  you  conceive  of  a  cylinder,  its  boundaries,  its  surfaces? 

Can  you  form  geometrical  concepts  of  other  solids? 


POSTULATES. 


Can  any  two  points  in  the  same  plane  be  joined  by  a 
straight  line?  Can  you  think  of  any  two  points  not  in  the  same 
plane?  Can  you  think  of  any  three  points  not  in  the  same 
plane? 


PRELIMINARY.  13 

Is  it  self-evident  that  any  straight  line  may  be  produced 
to  any  length  in  either  direction? 

May  a  circle  be  drawn  with  any  point  as  a  center  and  with 
any  finite  straight  line  as  a  radius? 

Can  a  figure  be  moved  unaltered  to  a  new  position? 

Is  it  possible  to  think  of  two  equal  geometric  cubes  being 
so  placed  that  they  will  coincide? 

Can  you  state  five  postulates? 


AXIOMS. 


What  is  an  axiom? 

Give  conclusion  in  the  following  examples  and  state  the 
axiom  applicable. 

1.  Tom  and  John  are  each  the  same  age  as  I;  therefore — 

2.  I  have  the  same  amount  of  money  as  Brown  or  Smith; 
therefore — 

3.  A  =  B  and  C  =r  B;  therefore— 

4.  Brown  has  as   much  money  as  Smith,  and  Jones  as 
Robinson;  .*.  Brown  and  Jones  together  have — 

A=:Band  C  =  D;  .-.  A  +  C=? 

5.  Two  armies  are  equal  in  number,  each  loses  500  men 
in  battle;  consequently — 

6.  Brown  and  Smith  are  each  double  the  height  of  the 
dwarf  Jones;  .*.  — 

7.  M  is  9  times  N,  R  is  also  9  times  N;  /.  — 

8.  A  is  J  of  B,  C  is  i  of  B;  .'.  — 

9.  Xis  §  ofY,  ZisfofY;  .-.— 


14  GEOMETRY. 

10.  My  whole  hand  is  larger  thau  my  thumb.  State 
axiom. 

11.  One-third  of  an  apple  is  less  than  the  whole  of  it. 
State  axiom. 

12.  I  am  older  than  you.  In  5  years  I  shall  still  be  older. 
State  axiom. 

I'd.  Smith  has  less  money  than  Jones,  each  spends  $5. 
Draw  conclusion  and  state  axiom. 

14.  If,  when  a  sheet  of  paper  is  placed  on  another,  their 
edges  exactly  coincide —      State  conclusion  and  axiom. 

15.  If,  when  one  line  is  placed  on  another,  their  ex- 
tremities coincide  and  every  point  in  the  first  line  coincides 
with  a  corresponding  point  in  the  second  line,  the  lines  are 
equal.     State  axiom. 

16.  Is  it  possible  for  the  extremities  of  two  straight  lines 
to  coincide  when  the  other  parts  of  the  lines  do  not  coincide? 

17.  Can  you  state  the  axiom  which  your  answer  suggests? 

18.  How  does  the  carpenter  get  a  straight  line  between 
two  points  without  using  a  straight-edge?  What  is  the  short- 
est distance  between  any  two  points?  Do  you  think  your 
answer  self-evident? 

19.  How  many  points  are  necessary  to  determine  the 
direction  of  a  straight  line?  How  many  straight  lines  can  be 
drawn  between  the  same  two  points? 

20.  In  how  many  points  can  two  straight  lines  intersect? 
Why?     Can  you  give  an  axiom  for  your  answer? 


DEFINITIONS. 

It  is  of  vital  importance  that  the  pupil  shall  be  able  to 
give  clear,  exact  definitions  to  all  terms  used  in  Geometry.  Of 
course,  he  must  fully  understand,  and  be  prepared  to  illustrate 
any  definition  given.  The  questions  previously  given  were 
designed  to  lead  the  pupil  so  far  as  possible  to  formulate  his 
own  definitions.     But  many  of  the  terms  in  Geometry  are  diflS- 


DEFINITIONS.  15 

cult  to  define,  and  the  pupil  can  compare  his  definitions  with 
those  here  given. 

Upon  these  definitions  and  upon  the  axioms  and  post- 
ulates rest  the  demonstrations  of  the  truths  of  Geometry. 
But  do  not  mistake  the  mere  learning  of  these  truths  to  be  the 
object  of  the  study.  //  is  the  ability  to  reason  which  we  acquire — 
their  demonstration. 


A  solid  is  a  limited  portion  of  space.  Its  dimensions  are 
length,  breadth,  and  thickness. 

The  pupil  can  conceive  space  to  be  divided  into  a  multi- 
tude of  forms  or  shapes.  Each  form  pictured  is  a  solid.  The 
geometrical  solid  contains  no  matter;  it  is  the  limited  portion 
of  space  conceived  by  the  nmid. 


A  surface  is  the  boundary  of  a  solid.  It  divides  space  into 
parts  and  can  be  conceived  without  the  solid,  so  the  definition 
is  often  given:  A  surface  is  thai  which  has  length  and  breadth 
without  thickness. 

(1)  K plane  surface  or  plane  is  a  surface  in  which  Many 
two  points  are  joined  by  a  straight  line,  every  point  in  the  line 
will  lie  in  the  surface. 

(2)  A  surface,  no  part  of  which  is  plane,  is  called  a 
curved  surface. 


A  line  is  the  boundary  of  a  surface.  We  can  conceive  the 
line  without  the  surface  and  define  it  to  be  that  which  has 
length,  but  7ieither  breadth  nor  thickyiess. 

(1)  A  straight  line  has  the  same  direction  throughout 
its  entire  length. 


16  GEOMETRY. 

(2)  A  curved  line  is  a  line  no  part  of  which  is  straight 
Hereafter  the  term  /ine  will  be  understood  to  mean  straight 
line,  and  curve  to  mean  curved  Hfie. 

(3)  Parallel  lines  are  everywhere   equidistant,  or   lines 
which  will  never  meet,  no  matter  how  far  they  are  produced. 


A  i)oint  is  the  extremity  of  a  line.  It  is  also  defined  as 
that  which  has  no  dimension,  but  position  only. 

We  may  further  explain  points,  lines,  and  surfaces  as 
follows: 

(1)  The  simplest  concept  that  can  be  formed  is  of  a 
point  which  has  no  magnitude. 

(2)  A  line  is  described  by  a  moving  point.  When  the 
point  does  not  change  its  direction,  a  straight  line  is  described; 
when  the  point  constantly  changes  its  direction,  a  curved  line  is 
described. 

(3)  A  surface  may  be  described  by  a  moving  line.  [How 
may  a  line  move  and  not  form  a  surface?] 

(4)  When  a  surface  is  moved  in  a  certain  manner,  a 
solid  is  generated  [How  may  a  plane  surface  be  moved 
without  generating  a  solid  ?] 


Solids,  surfaces,  and  lines  are  called  geometrical  magni- 
tudes. 

6. 

Geometry  is  the  science  which  treats  of  geometrical  con- 
cepts. 

7. 

A  geometrical  figure  is  any  combination  of  points,  lines, 
and  surfaces. 


DEFINITIONS.  17 

8. 

A  plane  figure  is  one  in  which  all  the  points  and  lines  lie 
in  the  same  plane. 

9. 

Plane  Geometry  treats  oi  plane  figures. 

10. 

Solid  Geometry  treats  of  figures  all  of  which  are  not  in 
the  same  plane. 

11. 

A  plane  angle  is  the  differefice  in  direction  of  two  lines 
which  m^et  or  nright  meet. 

Thus  A  B  C,  or  C  B  A,  is  an  a^gle  with  sides,  B  C  and 
B  A,  and  vertex  at  B.     The  size  of  the  angle  depends  upon 


the  amount  of  divergence  of  the  lines,  and  not  upon  their 
length. 

(1)  When  the  lines  point  in  exactly  opposite  directions, 
the  angle  is  called  a  straight  ajigle.  If  two  lines  be  drawn 
from  B,  as  in  the  figure,  two  angles  are  formed,  each  less  than 
four  right  angles.  Of  these  angles  the  smaller  is  always  un- 
derstood if  "the  angle  at  B"  is  mentioned,  unless  it  is  other- 
wise stated.  The  two  angles  at  B  having  the  same  sides  are 
called  conjugate  angles. 

Suppose  the  line  B  A  continues  to  revolve  about  the  ver- 


18  GEOMETRY. 

tex  B  until  it  passes  a  straight  angle  and  comes  around  to  the 
line  B  C,  or  forms  two  straight  angles,  a  perigon  is  formed. 
Angles  are  usually  measured  in  degrees,  minutes,  and  sec- 
onds. A  perigon  contains  360  degrees.  A  right  angle  is 
one-half  of  a  straight  angle. 

(2)  The  lines  which  form  a  right  angle  are  said  to  be 
perpendicular  to  each  other. 

(3)  A  straight  angle  equals  two  right  angles.  An  acute 
angle  is  less  than  a  right  angle.  An  obtuse  angle  is  greater 
than  a  right  angle  and  less  than  a  straight  angle. 

(4)  If  an  angle  is  greater  than  a  straight  angle,  and  less 
than  a  perigon,  it  is  said  to  be  reflex. 

(5)  Oblique  angles  are  either  acute  or  obtuse;  and  oblique 
li7ies  are  those  which  are  not  perpendicular  to  each  other. 

(6)  The  point  where  the  lines  which  form  the  sides  of 
an  angle  meet  is  called  the  vertex. 

(7)  When  two  angles  have  the  same  vertex  ana  a  com- 
mon side,  they  are  called  adjacent  angles. 

(a)  Draw  two  adjacent  angles,  both  of  which  are  acute. 
Can  their  sum  be  a  straight  angle?  an  obtuse  angle?  an 
acute  angle  ?  a  right  angle  ? 

(b)  Draw  two  right  angles  which  are  adjacent.  What 
is  their  sum  ? 

(c)  Draw  two  right  angles  which  have  the  same  vertex, 
but  are  not  adjacent. 

(d)  Draw  an  obtuse  angle  and  a  right  angle  which  are 
adjacent;  compare  their  sum  with  a  straight  angle. 

(8)  When  the  sum  of  two  angles  is  a  right  angle,  each  is 
called  the  complement  of  the  other,  and  they  are  called  comple- 
mentary angles.  Draw  two  complementary  angles  (1)  that  are 
also  adjacent  angles;  (2)  that  are  not  adjacent  angles. 

(9)  When  the  sum  of  two  angles  is  a  straight  angle,  each 
is  said  to  be  the  supplement  of  the  other,  and  the  angles  are 
called  supplementary  angles;  or  supplementary  angles  are 
angles  whose  sum  is  a  straight  angle. 


DEFINITIONS.  19 

(a)     Draw  two  supplementary  adjacent  angles. 

{b)  Draw  two  supplementary  angles  which  are  not  adja- 
cent, but  yet  have  the  same  vertex. 

(c)  Can  you  draw  two  supplementary  angles,  (1)  when 
both  are  acntef  (2)  when  both  are  obtuse?  (3)  when  one  is 
a  right  angle  and  the  other  acutef  (4)  when  one  is  obtuse 
and  the  other  acute? 

id)  What  are  the  conditions  under  which  two  angles 
may  be  supplementary? 

(10)  When  two  lines  intersect,  the  opposite  angles  are 
said  to  be  vertical  angles. 


Lines  A  B  and  C  D  intersect  at  E,  forming  angles  a,  b,  c,  d. 

{a)  Name  the  pairs  of  vertical  angles  ?  Do  they  appear 
to  be  equal?  to  be  complementary?  to  be  supplementary? 

{b)  Select  four  pairs  of  supplementary  angles.  Tell 
why  they  are  supplementary  ayigles. 

(c)     Name  two  straight  angles. 


12. 

A  proposition  is  a  statement  of  somethijig  to  be  considered 
or  to  be  done. 

(1)  A   theorem  is   a  proposition   stating    a  geometrical 
truth. 

(2)  K  problem  is  a  proposition  requiring  something  to 
be  done. 


20  GEOMETRY. 

(3)  An  axiom  is  a  theorem  so  elementary  that  no  proof  is 
required. 

It  is  self-evident  to  those  who  understand  the  terms  used 
in  expressing  it. 

(4)  A  postulate  is  a  problem  so  simple  that  its  con- 
struction is  admitted  as  possible  to  be  done. 

(5)  A  corollary  is  a  theorem  whose  truth  is  easily  de- 
duced from  a  preceding  proposition. 

(6)  A  scholum  is  a  remark  upon  a  preceding  proposition. 

13. 

A  theorem  may  be  divided  into  two  parts: 

(1)  The  hypothesis,  which  gives  the  data,  or  the  facts 
admitted  to  be  true. 

(2)  The  conclusion,  or  that  which  we  wish  to  prove  must 
follow  from  the  facts  admitted  by  the  hypothesis;  e.  g.\ 

If  two  triangles  have  two  sides  and  the  included 
angle  of  the  one  equal,  respectively,  to  two  sides  and 
the  included  angle  of  the  other,  the  triangles  are 
equal. 

What  is  known,  or  admitted  to  be  true,  about  the  two 
triangles  above  mentioned?  What  are  we  required  to />r^z^^ 
is  true'^. 

Or,  if  M  N  is  a  perpen- 
dicular intersecting  A  B  at 
E,  then  M  N  is  the  only  li?ie 
that  can  be  drawn  perpendic-      ^ 
ular  to  A  B  at  E. 

What  is  granted  to  be 
true?  What  are  we  re- 
quired to  prove  ? 


AXIOMS  AND  POSTULATES.  21 

AXIOMS. 
14. 

(1)  Things  equal  to  the  same  thing,  or  equal  things,  are 
equal  to  each  other. 

(2)  If  the  same  operation  be  performed  on  equals,  the 
results  will  be  equal. 

(3)  The  whole  is  greater  than  any  of  its  parts. 

(4)  The  whole  is  equal  to  the  sum  of  all  its  parts. 

(5)  If  equals  are  added  to  unequals,  the  sums  will  be 
unequal  in  the  same  sense. 

(6)  If  equals  are  subtracted  from  unequals,  the  remain- 
ders will  be  unequal  in  the  same  sense. 

(7)  Equals  may  be  substituted  for  equals. 

(8)  Magnitudes  whose  boundaries  coincide  are   equal. 

(9)  Two  points  determine  but  one  straight  line. 

(10)  Two  straight  lines  can  intersect  in  but  one  point. 

(11)  A  straight  line  is  the  shortest  distance  between  two 
points. 

(12)  Through  the  same  point  but  one  line  can  be  drawn 
parallel  to  a  given  line. 

POSTULATES. 

15. 

Let  it  be  granted: 

(1)  That  a  straight  line  can  be  drawn  joining  any  two 
given  points. 

(2)  That  a  straight  line  can  be  produced  to  any  extent 
in  either  direction. 

(3)  That  a  circle  can  be  drawn  with  any  point  as  the 
center  and  any  finite  straight  line  the  radius. 

(4)  That  on  the  greater  of  any  two  lines  can  be  cut  off  a 
line  equal  to  the  less. 

(5)  That  a  figure  can   be   moved   unaltered   to   a  new 
position. 

(6)  That  two  equal  magnitudes  can  be  made  to  coincide- 


22  GEOMETRY. 


SUGGESTIONS  TO  PUPILS.     ' 

The  first  step  in  the  solution  of  a  geometrical  problem  is 
to  study  it  very  carefully  to  understand  the  meaning  of  the 
language.  The  second  step  is  the  careful  construction  of  a 
figure  which  shall  afford  a  clear  conception  of  what  we  have 
to  do  or  prove.  By  constructing  your  figure  carefully,  rela- 
tions between  lines,  angles,  etc.,  are  often  suggested  which 
might  otherwise  escape  attention  if  the  figure  were  carelessly 
constructed.  Neatness  is  conducive  to  accuracy,  while  care- 
lessness tends  to  inaccuracy.  If  your  figure  suggests  certain 
relations,  you  are  now  ready  to  satisfy  yourself  whether  they 
are  real  or  apparent. 

Your  success  and  progress  in  solving  geometrical  prob- 
lems will  depend  on  your  habit  of  watching  for  new  properties 
that  present  themselves  in  various  ways.  The  construction 
of  a  figure  should  be  such  that  it  shall  not  exhibit  apparent 
relations  not  involved  in  the  problem  illustrated.  That  is, 
lines  should  not  seem  equal,  or  to  be  at  right  angles  when 
they  are  not  necessarily  so.  Triangles  should  not  seem  to  be 
isosceles  or  right-angled  unless  the  conditions  of  the  problem 
require  it.  It  is  better  to  make  a  triangle  whose  angles  are 
about  75°,  45°,  60°,  for  illustrating  in  general. 

If  quadrilaterals  are  spoken  of  in  a  problem,  use  the 
trapezium,  and  not  the  parallelogram.  It  is  a  good  plan  to 
draw  the  figure  of  the  problem  in  heavy  lines,  and  those  used 
as  helping  lines  more  lightly,  or  in  dotted  or  broken  lines. 
Always  letter  every  point  of  the  figure  which  may  be  referred 
to  as  you  proceed  with  your  discussion.  Keep  the  same  figure 
as  long  as  possible.  Drawing  new  figures  may  distract  the 
attention  from  a  course  of  reasoning.  After  having  exhausted 
the  properties  of  the  given  figure,  auxiliary  lines  may  be  drawn 
and  resulting  properties  noted.  The  most  useful  auxiliary 
lines  are  obtained  by — 

(1)     Joining  two  given  points. 


SUGGESTIONS  TO  PUPILS.  28 

(2)  Drawing  a  line  through  a  given  point  parallel  to  a 
given  line. 

(3)  Drawing  a  line  perpendicular  to  a  given  line  at  a 
given  point  within  the  line,  or  from  a  given  point  without  the 
line. 

(4)  Producing  a  line  its  own  length,  or  the  length  of 
another  given  line. 

In  preparing  your  lessons,  write  the  statement  of  the 
proposition  very  carefully,  as  you  have  worked  it  out,  to  bring  to 
class.     After  it  has  been  corrected,  then  write  it  in  your  book. 


24  PLANE  GEOMETRY. 


PI.ANK  GEOMETRY. 


16. 


The  demonstration  or  proof  of  a  theorem  must  be  based 
upon  definitions,  axioms,  postulates,  and  previously  proved 
theorems.  One  of  the  simplest  methods  of  proof  of  the  equal- 
ity of  two  figures  is  to  show  that  when  one  is  superposed  upon 
the  other  their  boundaries  coincide,  and  the  figures  are  con- 
sequently equal,  by  Axiom  8. 

17. 

Thus— suppose  we  wish  to  prove  that 

All  straight  angles  are  equal. 

What  is  a  straight  angle?  We  know  what  the  definitioji 
says,  nothing  more.  The  pupil  must  here  review  the  definition 
until  there  is  no  doubt  in  his  mind  about  what  it  states,  and  he  can 
fully  illustrate  it. 

(1)  What  is  an  angle  in  gener'al  ? 

(2)  What  is  a  straight  angle  ? 

After  the  definition  has  been  mastered,  let  him  draw  two 
straight  angles  and  attempt  to  prove  them  equal  by  showing 
that  their  sides  must  coincide  when  one  is  placed  upon  the 
other.  Write  the  authority  for  each  step  in  brackets  after 
each  statement.  Compare  your  proof  with  that  given  below 
and  see  if  yours  fails  in  any  essential  point. 


ANGLES.  25 

Theorem.     All  straight  angles  are  equal. 


Given:     A  E  B  and  M  N  O,  any  two  straight  angles. 
Required:     To    prove   that    angle   A  E  B    equals   angle 
M  N  O. 
Proof: 

(1)  Superpose  Z  A  E  B  upon  Z  M  N  O  so  that  point 
E  will  fall  upon  point  N  and  side  E  B  will  take  the  direction 
and  coincide  with  N  O.  [§15,  Post.  5 — Any  figure  can  be 
moved  unaltered  to  a  new  position.] 

(2)  Side  E  A  will  take  the  direction  of  side  N  M.  [§1 1 , 1— 
A  straight  angle  is  an  angle  whose  sides  point  in  exactly  oppo- 
site directions.] 

(3)  .-.  Z  A  E  B  =  Z  M  N  O.  [§14,  8— Magnitudes 
which  can  be  made  to  coincide  are  equal.]  But  Z  A  E  B  and 
Z  M  N  O  were  given  ayiy  two  straight  angles ;  . ' .  we  con- 
clude that  all  straight  angles  are  equal. 

18. 

Cor,     All  right  angles  are  equal. 

(The  proof  follows  directly  from  the  definition  of  a  right 
angle.) 

The  sections  and  exercises  are  numbered  consecutively 
throughout  the  entire  book. 

N.  B. —  The  proof  of  every  secHo7i  and  exercise  in  the  book  is 
required  of  the  pupil.  When  considered  too  difficult  for  the 
average  pupil,  partial  proofs  and  suggestions  are  given  to 
assist  him.  But  he  should  never  refer  to  these  hints  unless  he 
has  first  exhausted  his  own  resources  to  discover  a  proof  of 
his  0W71. 


26 


PLANE  GEOMETRY. 


TRIANGLES. 


19. 


Def.  A  triangle  is  a  portion  of  a  plane  bounded  by  three 
straight  lines.  Each  triangle  has  three  sides  and  three  angles. 
The  vertices  of  the  angles  of  the  triangle  are  called  the  vertices 
of  the  triangle.     The  sum  of  all  the  sides  is  called  the  perimeter. 

20. 

Triangles  are  classified  in  two  ways;  viz.,  (1)  with  respect 
to  sides;  (2)  with  respect  to  angles. 


21. 

The  Scalefie  triangle  has  no  sides  equal;  the  Isosceles 
triangle  has  two  sides  equal;  and  the  Equilateral  triangle  has 
all  sides  equal. 


Equilateral. 


TRIANGLES. 
22. 


27 


A  triangle  is  called  Acute  when  all  angles  are  acute;  Ob- 
tuse when  one  angle  is  obtuse;  and  Right  when  one  angle  is  a 
right  angle. 


Right. 

(1)  Draw  an  obtuse-isosceles  triangle. 

(2)  Draw  a  right-isosceles  triangle. 

(3)  Is  an  equilateral  triangle  acute? 

23. 

In  the  scalene  triangle  ABC,  produce  A  B  to  E,  then 
angle  C  B  E  is  called  an  exterior  angle. 

(1)  Form  the  exterior  angle  by  producing  {a)  A  C, 
{b)  B  C,  (d)  B  A,  {c)  C  A,  (<?)  C  B.   Redraw  the  triangle  each  time. 

Define  an  exterior  angle. 

(2)  Draw  a  triangle  and  then  draw  the  exterior  angle, 
which  shall  equal  the  adjacent  interior  angle.  Classify  the 
triangle. 

(3)  Draw  a  triangle  in  which  one  exterior  angle  is  acute. 
Classify  the  triangle. 

(4)  Which  classes  of  triangles  always  have  all  exterior 
angles  obtuse  angles? 


28  PLANE  GEOMETRY. 

24. 

The  base  of  a  triangle  is  the  side  upon  which  it  is  as- 
sumed to  stand.     Any  side  may  be  considered  the  base. 


25. 

The  angle  opposite  the  base  is  called  the  vertical  angle. 
Which  is  the  vertical  angle  of  triangle  ABC  when  A  B 
is  assumed  the  base?  when  B  C?  when  AC? 


26. 

The  vertex  of  the  vertical  angle  is  called  the  vertex  of  the 
triangle. 

27. 

The  altitude  of  a  triangle  is  the  perpendicular  distance 
from  the  vertex  of  the  triangle  to  its  base,  or  its  base 
produced. 

Can  you  draw  the  altitude  of  triangle  ABC  when  C  is 
the  vertex?  A?  B?  [Show  the  three  drawings.  Make  C 
an  obtuse  Z  ,  and  have  the  A  scalene.] 

Show  the  three  altitudes  of  a  right  triangle.  [  How 
many  are  drawn?] 

28. 

In  a  right  triangle  the  side  opposite  the  right  angle  is 
called  the  hypotenuse. 

Does  the  altitude  of  a  right  triangle  ever  fall  without  the 
base?  Which  side  is  assumed  the  base  when  the  altitude  falls 
within  the  base? 


TRIANGLES.  29 

29. 

Obtuse  triangles  and  acute  triangles  are  called  oblique 
triangles. 

30. 

The  three  lines  drawn  from  the  vertices  of  the  triangle  to 
the  middle  points  of  the  opposite  sides  are  called  the  medians 
of  the  triangle. 

Draw  triangle  ABC  and  draw  its  three  medians. 

31. 

The  three  lines  bisecting  the  angles  of  the  triangle  are 
called  the  bisectors  of  the  angles  of  the  triangle. 

In  what  classes  of  triangles  do  the  bisectors  of  the  angles 
and  medians  appear  to  be  the  same  lines?  Observe  closely 
these  triangles  and  state  what  you  discover.  What  do  you 
observe  about  the  intersection  of  the  three  mediaiis  of  any 
triangle?  the  three  bisectors  of  the  angles? 


30 


PLANE   GEOMETRY.      BOOK  I. 


BOOK    I. 


32. 


Proposition  I. 

When  are  two  angles  equal?  two  triangles?  any  two 
magnitudes? 

Draw  a  horizontal  line  and  with  compasses  cut  off  equal 
parts. 

Construct  a  triangle,  ABC,  making  length  of  A  B,  8,  of 
A  C,  6,  and  of  B  C,  4,  of  the  equal  parts. 


Then  construct  a  triangle,  M  N  O,  making  M  N  equal  to 
A  B,  angle  at  M  equal  to  angle  at  A,  and  side  M  O  =  A  C. 
Will  the  remaining  angles  and  side  of  A  M  N  O  be  equal  to 
corresponding  angles  and  side  of  A  A  B  C?  You  may  test 
the  accuracy  of  your  answer  with  compasses,  but  Prop.  I.  will 
state  a  general  truth  about  all  triangles,  and  the  proof  de- 


TRIANGLES.  31 

pends  upon  a  course  of  reasoning,  wherein  the  only  authority 
we  may  give  are  definitions,  axioms,  and  postulates. 

Make:  (1)  M  NT  =  A  B;  (2)  /  M  =  Z  A;  (3)  M  O  = 
A  C.     Then  draw  N  O. 

Clearly  fix  in  mind  the  parts  of  the  triangles  which  are 
known  to  be  equal.  Can  you  place  one  upon  the  other  in  such 
a  way  that  they  must  coincide?  By  what  authority  can  you 
do  this?  If  you  place  i\  M  N  O  upon  A  A  B  C,  upon  what 
point  of  A  A  B  C  will  you  place  point  M  ?  What  direction 
will  you  let  M  N  take  ?  Where  must  point  N  fall  ?  What 
axiom  proves  your  answer?  Will  you  fold  A  M  N  O  above 
A  B  or  below?  If  above,  will  M  O  take  the  direction  of 
AC?  By  what  axiom  ?  Will  O  fall  on  C  ?  Why  ?  Must  the 
line  N  O  ivholly  coincide  with  B  C  ?     Give  authority. 

Are  the  As  then  equal  ?     [Auth.] 

Again  review  the  sides  and  angles  that  were  known  to  be 
equal. 

DoesN   0==    BC?     [Auth.] 

Does  Z  N  =  Z   B  ?     [Auth.] 

Does  z  0=  Z  C?     [Auth.] 

Now  let  us  formally  prove  the  general  truth,  or  theorem, 
that  the  drawing  and  reasoning  lead  us  to  declare.  The  theorem 
makes  a  statement,  not  about  these  particular  triangles,  but 
about  any  two  triangles  in  which  the  given  conditiojis  are 
known  to  exist.  The  reasoning  does '  7iot  depend  upon  the 
mechanical  accuracy  of  the  drawing.  We  state  that  M  N  = 
A  B,  M  O  =  A  C  and  Z  M  =  Z  A,  not  because  they  were  so  con- 
structed (in  fact,  their  geometrical  construction  depends  upon 
problems  not  yet  studied),  but  because  they  are  so  giveyi  as  the 
conditions  upo?i  which  we  base  our  demonstration. 

The  order  of  arrangements  should  be  as  given  below: 

I.     General  enunciation  of  the  theorem. 

(1)  Hypothesis. 

(2)  Conclusion. 


32 


PLANE   GEOMETRY.      BOOK  I. 


II.  Particular  enunciation  of  the  figure  drawn. 

(1)  Hypothesis. 

(2)  Conclusion. 

III.  Proof. 

Note  carefully  the  steps  in  the  following  demonstration: 

Demonstration  of  Prop.  I.    (Model  for  Pupils.) 

/.  Theorem.  If  any  two  triangles  Have  two  sides 
and  the  included  angle  of  the  one  equal  respectively 
to  two  sides  and  the  included  angle  of  the  other,  the 
triangles  are  equal. 

J  I.     Given:     A  A  B  C  and  A  D  E  F,  in  which 

(1)  A  B  =  D  E, 

(2)  A  C  ==  D  F,  and 

(3)  z  B  A  C  =  Z  E  D  F. 
Required:      To  prove  that  AABC=ADEF. 
///.     Proof: 


(1)  Superpose  A  A  B  C  upon  A  D  E  F  so  that  point  A 
shall  fall  upon  point  D,  and  A  C  shall  take  the  direction  of 
D  F,  and  fold  A  A  B  C  above  the  line  D  F.  [§  15,  5,  Postu- 
late— A  figure  can  be  moved  unaltered  to  a  new  position.] 

(2)  A  C  ==  D  F;     [Hypothesis.] 

.  • .  C  will  fall  on  F.  [§15,6  (Post.  6)— Equal  magnitudes 
can  be  made  to  coincide.] 


TRIANGLES.  33 

(3)  Z  Ar=  Z  D;     [Hyp.] 

.-.  A  B  will  take  the  direction  of  D  E.     [§  15,  6.] 

(4)  A  B  =  D  E;     [  Hyp.] 

.-.  B  will  fall  upon  E.     [§15,  6.] 

(5)  C  falls  upon  F,  and  B  falls  upon  E;     [  P.  P.] 

.-.  B  C  coincides  with  E  F.  [§14,  9 — Two  points  deter- 
mine but  one  straight  line.] 

(6)  A  C  coincides  with  D  F,     [P.  P.] 

A  B  coincides  with  D  E,     [P.  P.]  and 
B  C  coincides  with  E  F;     [P.  P.] 
.-.  A  A  B  C  =  A  D  K  F.     [§  14,  8— Magnitudes  whose 
boundaries  coincide  are  equal  ]     Q.  E.  D. 

Let  the  pupil  carefully  prepare  written  proof  of  Prop  I.; 
use  the  above  figure,  but  assume  Z  s  B  and  E  and  the  includ- 
ing sides  respectively  equal. 


33. 


CIRCLES— Definitions. 

(1)  A  circle  is  a  plane  figure  bounded  by  a  curved  line 
every  point  of  which  is  equidistant  from  a  point  within,  called 
the  ceyiter. 

(2)  The  curved  line  which  bounds  the  circle  is  called  the 
circumference. 

(3)  Any  part  of  the  circumference  is  called  an  arc. 

(4)  A  straight  line  which  joins  the  ends  of  an  arc  is 
called  a  chord. 

(5)  A  radius  is  a  straight  line  which  joins  the  center  to 
any  point  in  the  circumference. 

(6)  The  longest  possible  chord  passes  through  the  cen- 
ter, and  is  called  the  diameter. 


34  PLANE   GEOMETRY.      BOOK  I. 

(7)  Theorem.  Circles  which  have  equal  radii,  or  equal 
diameters  are  equal;  and  conversely,  if  circles  are  equal,  they 
have  equal  radii  and  equal  diameters. 

Pupil  will  give  the  proof. 

[Hint. — Use  method  of  superposition.] 


Definitions— Homologous  Parts  of  Equai^  Figures. 

34. 

A  polygon  is  a  plane  figure  bounded  by  straight  lines. 

35. 

A  quadrilateral  is  a  polygon  having  four  sides. 

36. 

A  parallelogram  is   a   quadrilateral   having   its  opposite 
sides  parallel. 

37. 

A  rectangle  is  a  parallelogram  having  right  angles. 

38. 

A  square  is  a  rectangle  having  equal  sides. 

Review  Prop.  I.     (Quote  it.) 

Hereafter  we  can  prove  that  two  triangles  are  equal  by 
showing  that  two  sides  and  the  included  angle  of  the  one  are 


TRIANGLES. 


equal  respectively  to  two  sides  and  the  included  angle  of  the 
other,  and  then  quoting  Prop.  I.;  e.  g.,  if  we  have  given: 


(1)  The  square  A  B  C  D, 

(2)  A  M  =  B  N, 

we  can  prove  AADM=ABCN; 

(1)  -.-ADrirBC,     [?] 

(2)  Z  A  =  Z  B.     [?]  and 

(3)  AM=:BN;     [Hyp.] 

(4)  .• .  A  A  D  M  =  A  B  C  N.     [ Prop.  L] 

It  is  not  necessary  to  superpose  again  and  show  that  the 
triangles  must  coincide.  That  would  be  re-proving  Prop.  I., 
which  is  a  previously  proved  i)roposition  (note  abbreviation 
"P.  P.")  and  can  be  cited  as  authority  just  as  we  cite  axioms, 
definitions,  and  postulates. 

So  we  have  proved  the  above  triangles  equal,  using  only 
four  steps,  but  we  do  not  know  which  angles  correspond  and 
are  equal;  we  must  not  say  angle  at  D  equals  angle  at  C 
because  it  appears  to  be  true.  The  reasoning  does  not  at 
all  depend  upon  the  appearance  of  figures.  It  is  possible 
for  the  sides  to  be  so  nearly  equal  that  we  could  not  tell 
whether  Z  at  D  equaled  angle  at  C  or  at  N ;  furthermore  the 
drawing  might  be  inaccurate  and  not  agree  with  the  condi- 


36  PLANE   GEOMETRY.      BOOK  I. 

tions  [hypothesis],  and  the  appearance  would  greatly  mislead 
us.  We  look  for  the  equal  angle  by  first  looking  for  the  sides 
which  are  given  equal,  h  M  is  given  equal  to  B  N,  and  the 
ayigles  opposite  these  equal  sides  are  equal;  . ' .  in  As  ADM 
and  B  C  N  Z  at  D  =  Z  at  C. 

Also  A  D  was  proved  equal  to  B  C ;  hence  the  angles  op- 
posite these  equal  sides  are  equal ;  . ' .  Z   M  ^  Z  N. 

In  the  same  way  we  can  prove  D  M  ^  C  N,  being  oppo- 
site the  equal  angles  A  and  D  respectively. 

We  have  the  definition  given  below,  in  §  39. 

39. 

Def.  In  equal  figures,  equal  sides  are  opposite  the  angles 
which  are  known  to  be  equal,  and  equal  angles  are  opposite  the 
sides  which  are  known  to  be  equal.  Or,  in  equal  figures,  the 
homologous  parts  are  equal.  Homologous  sides  are  opposite 
equal  angles  and  homologous  angles  are  opposite  equal  sides. 


TRIANGIyKS— Exercises. 

The  exercises  in  this  book  are  given  for  two  reasons: 
(I)  To  give  the  pupil  facility  in  making  deductions  from 
data  given.  (2)  To  afford  abundant  application  of  this  pre- 
viously proved  proposition.  I^et  the  pupil  draw  just  so  much 
of  the  figure  as  is  required  in  each  exercise.  In  Ex.  2  the 
pupil  who  has  mastered  Prop.  I.  and  who  fully  understands 
the  discussion  o{  homologous  parts  of  equal  figures  will  be  able 
to  prove  many  triangles,  angles  and  sides  equal.  An  exercise 
upon  a  previously  proved  proposition  is  called  a  rider.  Let 
the  pupil  see  how  many  riders  he  can  make  by  answering  the 
questions  asked  in  Ex.  2. 


TRIANGLES. 


37 


KXERCISES. 


1.  Given  a  square  and  one  of  its  diagonals,  what  can 
you  prove? 

2.  Given  the  square  A  B  C  D  and  the  arc  E  O  F  with 
radius  C  E  and  center  C,  E  and  F  being  points  in  the  lines 
C  D  and  C  B,  respectively,  and  O  being  in  the  diagonal  A  C. 
Draw  F  A  and  E  A.  (I)  How  many  lines,  Zs,  and  As  can 
you  prove  equal?  (2)  By  joining  fixed  points,  how  many  pairs 
of  /\s  can  you  prove  equal? 


40. 

Proposition  II. 


Draw  2  oblique  As,  A  B  C  and  D  E  F,  making  A  B  =  D  E, 
Z  D  =r  Z  A,  and  Z  E  ==  Z  B. 

Can  you  prove  these  As  equal?  Use  method  of  super- 
position.    Write  Prop.  II. 

[yT//^/.— Superpose  AD  E  F  upon  AA  B  C  so  that  D  will 
fall  upon  A  and  D  E  will  take  the  direction  of  A  B.  Must  E 
fall  upon  B?     Can  you   now  show  that  F  must  fall  upon  C? 


38 


PLANE  GEOMETRY.      BOOK  I. 


Must  D  F  take  the  direction  of  A  C?  Why  must  F  fall  upon 
A  C  or  A  C  produced?  Must  E  F  take  the  direction  of  B  C? 
Must  F  also  fall  upon  B  C  or  B  C  produced?  Must  F  then 
fall  upon  two  lines  or  those  lines  produced?  What  point  is 
common  to  A  C  and  B  C?  How  many  points  in  common  is  it 
possible  for  two  lines  to  have?] 

Exercises. 


8.  Given  the  square  1  2  3  4,  1  5  =r  6  2,  and  Z  5'  =  /  6'. 
What  conclusions  can  you  draw  ? 

4.  Draw  diagonals.  Using  what  you  have  just  proved, 
what  new  As  can  you  prove  equal?  What  new  lines  and 
angles  can  you  prove  equal? 

5.  Join  5  and  4,  6  and  3.  What  new  As  can  you  prove 
equal?     What  new  lines  and  angles  can  you  prove  equal? 

6.  Draw  an  isosceles  A,  A  B  C,  with  A  B  and  A  C  the 
equal  sides.  Lay  off  B  D  on  B  A,  and  lay  off  C  K  =  B  D  on 
C  A.  Draw  C  D  and  B  K.  {a)  Why  is  A  D  =  A  K?  {b)  What 
Z  in  A  A  B  K  is  equal  to  an  Z  in  A  A  C  D?  Draw  separately 
the  pairs  of  As  which  appear  equal,  placing  them  in  similar 
positions,  {c)  In  As  A  B  K  and  A  C  D,  what  pairs  of  sides  are 
equal?  {d)  Why  is  A  A  B  K  equal  to  A  A  C  D?  {e)  What 
homologous  or  corresponding  parts  are  equal  as  a  result? 
(/)  Can  you  prove  other  pairs  of  As  equal? 


TRIANGLES.  39 

41. 

Proposition  III. 

What  have  you  learned  about  the  base  angles  of  any 
isosceles  /\? 

Now  we  wish  to  prove  this  truth  by  Demonstrative 
Geometry. 

Draw  any  isosceles  A ,  A  B  C,  making  A  B  and  A  C  the  equal 
sides.  Produce  A  B  and  A  C.  Measure  off  on  A  B  produced 
B  E,  and  on  A  C  produced  C  D  equal  to  B  E.  Join  points  E 
and  C.  What  2  new  As  have  been  formed?  Join  B  and  D. 
What  other  2  new  As  have  been  formed  which  appear  to  be 
respectively  similar  and  equal  to  the  first  2  As? 

Can  you  prove  the  pair  of  larger  As  equal? 

[ffi?iL — What  Z  is  common  to  both  As?  How  does  A  E 
compare  with  A  D?  Why?  If  you  are  still  unable  to  prove 
the  As  equal,  redraw  them,  placing  them  in  similar  positions.] 

Note  carefully  the  sides  and  Z  s  in  the  pair  of  smaller  As 
which  are  common  to  the  larger  As.  Now  prove  the  pair  of 
smaller  As  equal  in  all  their  parts. 

What  have  you  proved  about  Z  A  B  D  and  Z  A  C  E? 

What  have  you  proved  about  Z  D  B  C  and  Z  E  C  B? 

What  axiom  can  you  apply  to  show  that  Z  A  B  C  equals 
Z  A  C  B?     Write  a  general  statement  and  call  it  Prop.  III. 

Let  the  pupil  carefully  prepare  written  proof  of  Prop.  III. 
and  then  compare  with  the  demonstration  given  below. 

MODEI.   DEMONSTRATION. 

Theorem.  If  a  triangle  is  isosceles,  the  angles 
Opposite  the  equal  sides  are  equal. 

Given:     The  isosceles  A  A  B  C,  in  which  A  B  =  A  C. 
Required:     To  prove  that  Z  C  =  Z   B. 


40  PLANE   GEOMETRY.      BOOK  I. 

Proof:     Produce  the  equal  sides  A  B  and  A  C,  and  upon 
these  sides  produced  lay  off  the  equal  lines   B  E  and  C   D. 


Join  E  to  C  and  D  to  B,  forming  A  B   C  A  and  A  D  B  A, 
also  A  B  C  E  and  A  D  B  C. 

I.  Prove  AECA:=ADBA,  and  consequently  Z  ?«  = 
Z  n- 

(1)  A  B  =:  A  C,     [Hyp.] 

(2)  B  E  =  C   D;     [Construction.] 

(3)  .  • .  A  B  -f  B  E  =  A  C  +  C  D,  [  §  14,  2-If 
the  same  operation  be  performed  on  equals  the  results 
will  be  equal.] 

(4)  A  E  ==  A  D.  [§  14,  7— Equals  may  be  sub- 
stituted for  equals.] 

(5)  AC  =  AB,     [Hyp.] 

(6)  Z  A  is  common  to  both  triangles  ; 

(7)  .  • .  A  E  C  A  1=  A  D  B  A,  [§  32— If  two  trian- 
gles have  two  sides  and  the  included  angle,  etc.]  and 
/_  m  =^  /_  n.  [§  39 — Being  homologous  angles  of 
equal  As  opposite  the  equal  sides,  A  E  and  A  D.] 

II.  Prove  z\BCE=:ADBC,  and  consequently  Z  /> 

=    10. 

(1)  B  E  =  C  D,     [So  constructed.] 

(2)  E  C  =  D  B,  [§  39— Being  homologous  sides  of 
the  equal  triangles  ACE  and  A  B  D,  opposite  the 
common  angle  at  A.] 

(3)  Z  E  =  Z  D;  [g  39— Opposite  the  equal  sides 
A  D  and  A  E,  respectively.] 

(4)  .♦.  ABEC=  ABDC,  [§32.]  andZ;5>=:Z^. 
[§  39  —Being  opposite  the  equal  sides  D  C  and  B  E.] 


TRIANGLES. 


41 


III.      (1)      l_  m^  l_  n,     [P.  P.] 
CI)      L  o^  L   P;     [P.  P.] 

(3)     r  .m  —  o=^n  — />,  [§  14,  2.]  or  Z  C  =  Z  B. 
[§14.7.]  Q.E.  D. 

Exercises. 
7.     Can  you  prove  that  an  equilateral  A  is  equiangular? 

F  C 


Fig.  1. 


Fig.  2. 


8.  Given  the  square  A  B  C  D.  Draw  B  D,  and  make 
D  N  =  B  K.     Draw  arc  E  F  with  center  A  and  radius  A  E. 

Prove  (1)ZA  BD=ZA  D  B;  (2)  Z  A  K  N  =  Z  A  N  K; 
(3)   Z  A  E  F  =  Z  A  F  E. 

9.  In  the  second  figure,  A  B  is  diameter  of  0;  A  O  C  F 
is  a  square  erected  on  radius  A  O;  B  D  is  a  chord,  and  E  its 
middle  point.  Prove,  (1)  A  C  =  C  B ;  (2)  Z  O  A  C  =  Z  O  C  A ; 
(3)  Z  A  C  B=sum  of  the  Zs  C  A  B  and  C  B  A;  (4)ZOB  D 
=  ZODB;  (5)  The  sum  of  the  Z  sO  C  B  and  O  D  B=:Z  CBD. 

10.  Construct  A  B  C  an  equilateral  Z\.  and  A  B  D  an 
isosceles  A.  on  the  same  base,  A  B.  Prove  the  Z  C  A  D  = 
Z  C  B  D,  whether  the  As  are  on  the  same  side  or  on  opposite 
sides  of  A  B.  (1)  Join  D  and  C.  Produce  D  C,  if  necessary, 
until  A  B  is  cut.  Is  A  B  bisected?  Prove.  Make  further 
deductions. 

11.  \i  X  and  J  are  the  middle  points  of  the  equal  sides 
x\  B  and  A  C,  respectively,  of  the  isosceles  A  A  B  C,  prove  in 
two  ways  that  C  ;»;  =  B  >/.  Make  deductions.  E'  and  F  are 
points  in  the  base  B  C,  and  B  E  =  C  F.     Prove  A  E  =  A  F. 


42  PLANE  GEOMETRY.      BOOK  I. 

42. 
Proposition  IV. 

Draw  two  acute-angled  As,  ABC  and  D  E  F,  so  that  A  C 
==DF,  AB  =  DK,BC=KF.  Place  A^HFonAABC 
so  that  K  F  falls  on  B  C  and  vertex  D  falls  opposite  to  ver- 
tex A.  Join  A  and  D.  How  many  isosceles  As  are  formed? 
Can  you  prove  Z  D  equal  to  Z  A?      What  follows? 

Draw  again,  making  Z  s  at  K  and  B  obtuse  Z  s.  What 
follows?     Write  a  general  statement  and  call  it  Prop.  IV. 

43. 

Definition, — A  right  bisector  of  a  line  meets  it  at  right 
angles  and  bisects  it. 

44. 

Proposition  V. 

Draw  any  straight  line,  A  B,  and  fix  two  points,  P  and 
Q,  equidistant  from  the  ends  of  A.  B. 

Do  these  two  points  determine  the  right  bisector 
of  A  B.^ 

Draw  P  Q  and  if  necessary  produce  it  to  meet  A  B  at  C. 
Can  you  prove  the  angles  at  C  are  right  angles?  What  is  a 
straight  angle?  a  right  angle?  Can  you  prove  A  B  is  di- 
vided into  two  equal  parts?     Can  you  state  this  proposition? 

If  you  fail  to  state  Prop.  V.,  or  to  prove  it,  do  not  get  dis- 
couraged (the  proof  is  difficult  for  the  beginner),  but  carefully 
study  the  * 'hints"  given  below.  Master  each  step  before  read- 
ing the  next,  and  just  so  soon  as  you  discover  the  proof,  finish 
it  without  reading  further  until  after  you  have  carefully  pre- 
pared your  own  proof  in  full.  Ability  to  originate  a  single 
step  in  a  demonstration  will  give  you  power  with  which  to 
attack  future  demonstrations.     Struggle,  repeatedly  and  with 


TRIANGLES. 


43 


determination,  for  this  power  to  originate.  It  is  the  highest 
order  of  mental  achievement. 

[Hint.—l^^t  A  B  be  the  given  line  and  let  P  and  Q  be 
two  points  equidistant  from  the  ends.] 

To  prove  that  P  Q,  or  P  Q  produced,  is  the  rt.  bisector  of 
A  B. 


Let  P  Q  intersect  A  B  at  C.  Think  of  the  definition  of 
a  rt.  bisector.  We  must  prove  what  lines  are  equal  to  prove 
that  A  B  is  bisected  ?  We  have  learned  to  prove  lines  equal 
in  what  ways?  If  B  C  is  superposed  on  C  A,  can  we  prove 
that  the  lines  must  coincide  and  are  consequently  equal?  If 
not,  can  we  prove  B  C  and  C  A  similar  sides  oj  equal  figures  y 
and  are  consequently  equal  ?  Draw  two  lines  which  will 
form  two  triangles  of  which  B  C  and  C  A  are  sides,  respect- 
ively. Do  you  know  any  sides  and  angles  of  these  triangles 
equal?  Are  you  able  to  prove  the  triangles  equal  ?  (Think 
of  the  previous  propositions  you  have  proved,  and  just  what 
is  necessary  to  prove  the  triangles  equal  by  using  Prop.  I.,  or 
Prop.  II.,  or  Prop.  IV.)  If  you  do  not  know  sufficie7it  sides 
and  angles  of  these  triangles  equal  to  prove  the  triangles  equal 
by  either  of  the  previously  proved  propositions,  you  must 
form  other  triangles  and  try  to  prove  them  equal. 

(Note  well  just  what  was  lackifig  to  prove  the  above  tri- 
angles equal — was  it  not  the  angles  at  P,  or  at  Q?) 

If  you  need   the  angles   at   P   equal  in  order  to  prove 


44  PLANE  GEOMETRY.      BOOK  I. 

AACP  =  ABC  P,  by  joining  other  points  you  will  have 
two  new  triarigles  which  contain  these  angles  at  P.  Now 
if  you  can  prove  the  new  triangles,  A  P  Q  and  B  P  Q,  equal 
(redraw  figure  on  new  page  if  necessary),  you  can  then  prove 
angles  at  P  equal,  and  then  the  triangles  A  C  P  and  B  C  P 
equal,  and  finally  the  sides  B  C  and  C  A  equal. 

[The  pupil  must  understand  each  step  in  the  order  pre- 
sented. Review  until  you  clearly  see  just  why  each  step  is 
necessary?^ 

Then  B  C  =  C  A,  [P.  P.]  and  A  B  =  B  C  +  C  A; 
[Ax.  ?J 

.  • .  A  B  is  bivSected  by  P  Q,  and  P  Q  is  the  bisector  of 
AB. 

But  it  is  required  to  prove  that  points  P  and  Q  determine 
the  right  bisector;  hence  it  is  necessary  to  prove  angles  A  C  P 
and  B  C  P  right  angles.  What  are  right  angles  ?  (See  the 
definition.) 

What  is  the  sum  of  two  adjacent  angles  at  C? 

Can  you  prove  them  equal  ? 

Carefully  prove  Prop.  V.  Compare  with  proof  given  to 
Prop.  III.  if  neceSvSary. 

KXERCISKS. 

12.  Letter  the  intersection  of  A  D  and  B  C,  in  §42,  O. 
What  pairs  of  equal  As  in  the  figure?  Give  the  equal  homol- 
ogous or  corresponding  parts  resulting.  What  angles  at  O  are 
equal? 

13.  If  2  oblique  lines  are  drawn  from  the  same  point  in 
a  perpendicular  cutting  off  equal  distances  from  the  foot  of 
the  1,  how  are  these  two  lines  related? 


PROBLEMS.  45 

45. 

Proposition  VI. 

Problem.      To  bisect  a  given  straight  line. 

[What  proposition  treats  of  the  bisection  of  a  line?  What 
is  necessary  to  bisect  it?  How  can  you  find  the  necessary 
points?] 

Given:     A  B  a  straight  line. 

Required:     To  bisect  A  B. 

Construction:     [Let  the  pupil  construct  the  figure.] 

(1)  With  centers  A  and  B  and  any  radius  greater  than 
one-half  of  A  B,  describe  arcs  which  intersect  at  C  and  D. 

(2)  Draw  C  D,  cutting  A  B  at  E. 
Then  A  B  is  bisected  at  E. 

Proof: 

(1)     AC  =  BC,     [?] 

(-4)     AlsoAD  =  BD;     [?] 

(3)  .• .  A  B  is  bisected  by  C  D  at  E.     [  ? ] 

Q.  E.  F. 

46. 

Proposition  VII. 
Problem.      To  bisect  a  given  angle. 

[  Draw  an  angle  and  bisect  it.  The  construction  has  been 
learned  in  Constructional  Geometry,  but  it  is  now  required 
to  prove  that  the  angle  has  been  bisected.  You  are  required 
to  prove  angles  equal ;  review  the  propositions  in  which  this  is 
done.  By  joining  fixed  points,  construct  two  triangles  which 
contain  the  angles.  Prove  these  triangles  equal.  N.  B. — 
There  are  two  pairs  of  triangles  which  may  be  drawn.] 

Given:     The  angle  ABC. 

Required:     To  bisect  ABC. 

Construction:     [I^et  the  pupil  construct  the  figure] 

(1)     With  center  B  and  any  radius  less  than  B  A  or  B  C, 


46  PLANE   GEOMETRY.      BOOK  I. 

describe  an  arc  cutting  B  A  and  B  C  at  D  and  E  respectively. 

(2)  With  centers  D  and  E  and  any  radius  greater  than 
oue-half  of  D  E,  describe  arcs  which  intersect  at  F,  remote 
from  B. 

(3)  Draw  B  F,  forming  angles  A  B  F  and  C  B  F.  Thus 
A  B  C  is  bisected  by  B  F. 

Proof:     lycft  to  the  pupil. 

47. 

Proposition  VIII. 

Problem.  To  draw  a  perpendicular  to  a  given 
line  at  a  given  point  in  it. 

Giveyi:     (l)  The  line  A  B;  (2)  P,  a  point  in  A  B. 

Required:     To  draw  a  perpendicular  to  A  B  at  P. 

Construction:     Left  to  the  pupil. 

Proof:     Left  to  the  pupil. 

If  unable  to  construct  and  prove  the  above  problem,  con- 
sult the  *'hint"  below. 

[Flint. — What  proposition  enables  you  to  draw  a  perpen- 
dicular to  a  given  line?  But  at  which  point  in  the  line  does  it 
enable  you  to  draw  the  perpendicular?  Is  P  that  point?  If 
not,  cut  off  of  A  B  a  line  of  which  P  is  the  middle  point.  Be 
careful  to  fix  no  unnecessary  points.] 

48. 
Proposition  IX. 

Problem,  To  erect  a  perpendicular  to  a  line  from 
a  given  point  without  that  line. 

The  hypothesis,  construction,  and  proof  are  left  for  the 
pupil. 

If  unable  to  make  the  construction,  consult  the  "hint" 
below. 

[Hint. — Why  do  you  7wt  wish  to  draw  the  perpendicular 
to  the  mid-point  of  the  given  line?     But  it  is  07ily  by  having 


PROBLEMS.  47 

two  points  equidistant  from  the  ends  of  some  line  that  you  can 
draw  a  perpendicular.  How  then  can  you  cut  off  a  part  of 
the  given  line  of  which  P  is  equidistant  from  the  ends?  After 
cutting  off  this  part,  you  can  easily  fix  another  point  equi- 
distant from  its  ends;  and  thus  you  have  a  line  and  two  points 
equidistant  from  its  ends;  and  P  is  one  of  these  two  points.] 

Exercise. 

14.  Divide  a  line  into  four  equal  parts;  into  sixteen 
equal  parts. 

49. 

Proposition  X. 

Problem,     At  a  given  point  ifi  a  given  straight 
line  constrnct  an  angle  equal  to  a  given  angle. 
Left  to  the  pupil. 

Exercises. 

15.  Construct  an  isosceles  triangle:  (1)  When  base  and 
one  side  are  given;  (2)  when  the  angle  at  the  vertex  and  one 
side  are  given ;  (3)  when  an  angle  at  the  base  and  the  base  are 
given. 

16.  If  a  line  is  drawn  from  the  vertex  of  an  isosceles  A 
bisecting  the  base,  prove  that  it  is  1  to  the  base. 

17.  Construct  a  A  when  2  sides  and  the  included  Z  are 
given. 

18.  Construct  a  A  equal  to  a  given  A-  Show  three 
ways. 

19.  Let  A  B  C  be  an  isosceles  A,  A  B  =  A  C.  Let  bisector 
of  Z  B  meet  A  C  in  ;»;  and  the  bisector  of  Z  C  meet  A  B  in  y. 
Show  that  Bx  =  Oy. 

20.  If,  in  Fig  2,  Ex.  7,  a  point  in  the  circumference,  H, 
is  equidistant  from  B  and  D,  and  H  is  joined  to  O,  B,  and  D, 
what  As  are  equal?     What  lines  must  coincide? 


48  PLANE   GEOMETRY.      BOOK  I. 

21.  Draw  any  two  intersecting  Qs;  join  the  centers  and 
join  eacli  center  to  points  where  circumferences  intersect.  Can 
you  prove  the  equalityof  the  As  formed?  Can  you  form  other 
equal  As  by  joining  the  points  where  circumferences  intersect? 

{Suggestion, — Draw  Qs  in  all  possible  positions.  Seek 
for  exceptions  to  your  answers.] 

60. 

Proposition  XI. 

How  many  perpendiculars  may  be  erected  to  a 
given  line  at  a  given  point  in  that  line?  (The 
lines  are  assumed  to  be  in  the  same  plane ^ 

The  answer  at  first  seems  self-evident,  and  it  appears 
hardly  necessary  to  prove  that  there  can  be  but  one. 

All  the  proofs  given  hitherto  have  been  direct  proofs,  but 
here  we  use  an  indirect  method  of  proof.  We  suppose  that 
what  we  wish  to  prove  true  is  false  and  then  show  that  our 
supposition  leads  to  an  absurd  conclusion.  This  is  one  of  the 
indirect  methods  of  proof,  and  is  called  ''reductio  addbsurdum''' 
a  reduction  to  an  absurdity. 

[State  the  proposition.] 

Given:  (1)  the  line  A  B;  (2)  P,  a  point  in  A  B;  (3)  CPE, 
a  1  to  A  B  at  P.     [Pupil  draw  the  figure.] 

Required:  To  prove  that  C  P  B  is  the  only  1  that  can 
be  drawn  to  A  B  through  P. 

Proof:  Draw  any  other  line  through  P,  F  P  D,  and  sup- 
pose, if  possible,  that  F  P  D  is  another  1  to  A  B  at  P. 

[lyCt  the  pupil  draw  the  lines,  making  F  P  D  a  dotted  line 
and  let  C  and  F  be  above  the  line  A  B,  and  place  F  at  the 
right  of  C] 

(1)  :  Z  CPBisart.  Z;     [?] 

(2)  also,  Z  F  P  B  is  a  rt.  Z  ;    [?] 

(3)  .-.  Z  CPB=  Z  FPB.     [?1 


ANGLES.  49 

But  this  is  absurd  for  Z  C  P  B  is  greater  than  Z  F  P  B; 
[Ax.  ?] 

• .  *  the  supposition  that  F  P  D  was  another  L  led  us  to  an 
absurd  conclusion  (No.  3,  where  a  part  is  proved  equal  to  the 
whole),  we  must  reject  the  supposition  and  conclude  that 
F  P  D  is  not  a  1  to  A  B  at  P ;  and  since  F  P  D  is  any  other 
line  than  CPE  {the  true  1),  we  conclude  that  there  is  no  other 
1  and  that  C  P  E  is  the  only  1. 

[The  pupil  should  carefully  review  each  step  of  the  proof 
until  he  understands  the  full  force  of  the  reasoning] 


51. 

Proposition  XII. 

Draw  two  adjacent  angles.  How  many  lines  are  used  ? 
Draw,  using  three  lines,  two  lines. 

If  two  lines  meet  so  as  to  form  two  adjacent 
angles,  what  is  their  sum  compared  with  a  right 
angle? 

State  the  proposition  and  give  the  proof. 

[Hint.  —The  proof  is  not  difficult,  but  be  careful  not  to  use 
any  statement  which  is  not  given  in  previous  proposition,  or 
axiom,  or  definition.] 

52. 

Cor.  I.  What  is  the  sum  of  all  the  angles  that 
can  be  formed  on  the  same  side  of  a  given  straight 
line  at  a  given  point  in  that  line? 

State  and  prove. 


50  PLANE  GEOMETRY.      BOOK  I. 

53. 

Cor.  II.  What  is  the  sum  of  all  the  angles  that 
can  be  formed  in  the  same  plane  around  a  given 
point? 

State  the  corollary. 

Given:  (1)  Point  P;  (2)  angles  a,  b,  c,  d,  etc.,  formed 
around  P. 

Required:  To  prove  the  sum  of  angles  a,  b,  c,  d,  etc.,  is 
four  right  angles. 

Prool'  (The  pupil  should  fix  point  P  and  draw  any 
number  of  angles,  a,  b,  c,  d,  etc.,  so  that  any  side  produced 
will  not  coincide  with  any  other  side.  Then  produce  any  side, 
and  use  Cor.  1.) 

Questions. 

1.  Are  all  straight  angles  equal? 

2.  What  is  the  complement  of  an  angle?  the  supple- 
ment? 

3.  If  an  angle  is  double  its  complement,  what  fraction 
is  it  of  a  right  angle?  of  a  straight  angle? 

4.  If  an  angle  is  three  times  its  supplement,  what  frac- 
tion is  it  of  a  straight  angle?  of  a  right  angle? 

5.  What  are  supplementary  adjacent  angles?  How  large 
is  an  angle  whose  supplement  is  three  times  its  complement? 

6.  What  is  the  angle  between  the  bisectors  of  two  sup- 
plementary angles?  What  is  the  angle  between  the  bisectors 
of  two  complementary  adjacent  angles? 


ANGLES.  51 

7.  What  is  the  hypothesis  of  a  theorem?  the  conclusion? 
What  is  the  hypothesis  of  Prop.  III?  the  conclusion.  If  the 
hypothesis  and  conclusion  of  Prop.  Ill  were  interchanged, 
how  would  it  be  stated  ?     Do  you  think  it  is  true  ? 

54. 

Definition. —  The  converse  of  a  theorem  is  the  theorem  when 
the  hypothesis  and  conclusion  are  interchanged. 

65. 

Proposition  XIII. 

If  two  adjacent  angles  are  supplementary,  how 
are  their  exterior  sides  related? 

Stale  and  prove  Prop.  XIII. 

[fli?it. — What  is  a  straight  angle?] 

Is  Prop.  XIII  the  converse  of  any  proposition? 

66. 

Proposition  XIV. 

Given  the  isosceles  A  A  B  C  with  A  B  and  A  C  the  equal 
sides.  Join  A  with  the  middle  point  of  the  base  B  C.  Prove 
the  2  /\  s  formed  equal. 

How  does  the  median  meet  the  base?  Why? 
How  does  it  divide  the  triangle  ABC?  How  does 
it  divide  the  angle  at  the  vertex? 

Write  a  formal  statement  of  these  three  truths  and  call  it 
Prop.  XIV. 

Erect  a  1  at  the  mid-point  of  the  base  of  any  isosceles  A- 
Through  what  point  must  it  pass? 


52 


PliANE  GEOMETRY.     BOOK  I. 
57. 

Proposition  XV. 


Let  P  be  a  point  without  the  straight  line  A  B,  and  P  D 
be  a  1  to  it.  Suppose  it  is  possible  to  draw  another  1  from  P 
and  let  P  F  be  that  1.  Can  you  think  of  any  axiom  which 
this  supposition  violates? 

Produce  P  D  to  P'  making  D  P  =  D  P'.  Join  F  and  P'. 
Can  you  prove  F  P'  equal  to  F  P? 

If  P  F  is  1  to  A  B,  how  large  is  Z  x}  What  is  the  value 
of  Z  X  -^  I  x?  What  kind  of  a  line  is  P  F  P'?  But  what 
axiom  is  violated  if  our  supposition  is  true  ? 

Can  P  F  be  1  to  A  B? 

Can  any  other  line  than  P  D  be  drawn  from  P 
ItoAB?     Why? 

What  then  must  we  say  of  the  supposition  which  led  to  this 
absurd  conclusion? 

Write  a  formal  statement  of  the  truth  discovered  and  call 
it  Prop.  XV. 


ANGLES. 

58. 

Proposition  XVI. 


58 


L,et  the  two  straight  lines  O  P  and  M  N  intersect  at  I, 
forming  the  vertical  angles  x  and  x\  y  andy. 

What  is  the  value  of  Z  JK  +  L  x?     Quote  P.  P. 
What  is  the  value  of  Z  ^  +  Z  ^?     Quote  P.  P. 

Can  yoii  prove  that  /-  y  ^z  I-  y'} 

Prove  in  a  similar  manner  that  Ix  =  Ix' . 


Write  Prop.  XVI. 

EXERCISKS. 

22.  If  Z  y  and  Z  y  in  §  58  are  bisected,  prove  what  is 
true  of  the  bisectors. 

23.  If  Z  ^  and  Z  y  are  bisected,  prove  what  is  true  of 
the  bisectors.  What  can  you  say  of  the  bisector  of  Z  -^  pro- 
duced through  Z  x'} 

Surveyor's  Problem. 


.*>^** 


Suppose  a  surveyor  wishes  to  know  the  distance  between 
.A  and  B,  with  a  lake  between.     Suppose  that  the  distance 


54 


PLANE  GEOMETRY.     BOOK  I. 


A  B  is  not  over  300  feet  and  that  the  surrounding  land  is 
level;  tell  how  to  measure  the  distance  required  by  the  hint 
in  diagram,  and  prove  your  work. 


W 


It  is  required  to  find  the  distance  from  A  to  B,  with  a 
river  between.  Supposing  the  surrounding  land  to  be  level 
and  that  the  surveyor  has  a  transit  which  measures  angles  and 
which  enables  him  to  run  straight  lines;  can  you  tell  from  the 
suggestion  in  the  diagram  how  to  estimate  the  distance 
required  ? 


PERPENDICULARS.  56 

59. 

Proposition  XVII. 

Given  P  a  point  outside  of  the  line  A  B. 

What  is  the  shortest  line  that  can  be  drawn  from 
P  to  A  B? 

If  P  D  is  the  shortest  line,  how  should  it  be  drawn? 


[Try  to  state  and  prove  Prop.  XVII.  Use  the  above 
figure.] 

(If  you  fail,  consult  the  "hint"  given  below.) 

{Hint  —  Give7i:     P  D  a  1  to  A  B  at  D.] 

Required:     To  prove  P  D  the  shortest  line  from  P  to  A  B. 

Proof:  Draw  any  other  line,  P  F.  Produce  P  D  its  own 
length  to  P'.     Draw  P'  F. 

P  P'  <  P  F  P'.     [Auth.] 

P  D  <  P  F.     [Auth.     Pupil  must  prove  P'  F  —  P  F,  etc:\ 


PLANE  GEOMETRY.      BOOK  I. 


60. 

Proposition  XVIII. 


Given  the  right  As  A  B  C  and  DBF  having  the  hypote- 
nuse A  C  of  the  first  A  equal  to  the  hypotenuse  D  F  of  the 
second  A,  and  Z  jc  ==  Z  x' . 

■  Can  you  prove  the  As  equal  ? 

If  not,  superpose  A  D  H  F  upon  A  A  B  C  so  that  D  shall 
fall  on  A  and  D  F  take  the  direction  of  A  C.  Where  will  F 
fall?  Why?  What  direction  will  F  K  take?  Why?  Will  E 
fall  on  B?  W^hy?  If  not,  how  many  Is  would  be  drawn  from 
point  A  to  the  same  line? 

Suppose  Z  JV  =  Z  /  and  that  we  know  nothing  of  angles 
X  and  x';  prove  the  As  equal. 

Write  the  formal  statement  of  the  truth  proved  and  call  it 
Prop.  XVIII. 

61. 

Proposition  XIX. 


Given  the  right  As  A  B  C  and  D  K  F   with  A  C 
and  A  B  =  D  K. 


=rD  F 


TRIANGLES. 


57 


Can  you  prove  the  As  equal  ? 

Place  ADKFonAABCso  that  D  will  fall  on  A  and 
let  D  E  take  the  direction  of  A  B  and  let  F  fall  on  F'.  Must 
K  fall  on  B?  Why?  Will  C  B  and  F  E  form  one  straight  line? 
Why?  Will  C  A  F'  form  a  A?  Why?  Prove  Z  F  =  Z  C. 
Write  Prop.  XIX. 

62. 

Proposition  XX. 


Given:  (1)  Any  line^  2;  (2)  xB  Itoy  2  at  D;  (3)  P 
any  point  in  the  1  x  D;  (4)  P  B  and  P  A  lines  cutting  oflf  ofy  z 
the  unequal  distances  D  B  and  D  A,  D  B  being  less  than  D  A. 

Can  you  prove  PB  less  than  PA? 

Produce  PD  and  make  DQi=:DP.  Make  DB'=:DB. 
Join  Q  to  B'  and  to  A.  Produce  Q  B'  to  F.  Prove  B'  Q==B'P, 
and  A  Q  =:  A  P.  How  does  the  broken  line  P  B'  Q  compare 
with  P  D  Q?  Prove  P  F  B'  >  P  B'.  Prove  P  F  B'  Q  >  P  B'  Q. 
Can  you  prove  Q  A  P  >  Q  F  P? 

How  does  Q  A  P  compare  with  Q  B'  P? 

How  does  A  P  compare  with  B'  P? 

How  does  A  P  compare  with  B  P? 

Write  the  general  truth  proved  and   call    it  Prop.  XX. 


58  PLANE  GEOMETRY.      BOOK  I. 


Exercises. 


Given:     Construct  the  following  triangles: 

24.  Two  angles  and  the  included  side. 

25.  Two  sides  and  the  angle  opposite  one  of  them.  Dis- 
cuss, showing  under  what  conditions  the  construction  is 
possible. 

26.  Three  sides.     Discuss. 

27.  Hypotenuse  and  one  adjacent  angle  of  a  rt.  A« 
Discuss. 

28.  Twolegsof art.  A. 

29.  One  side  and  an  adjacent  acute  angle  of  a  rt.  A- 

30.  Hypotenuse  and  one  leg  of  a  rt.  A-     Discuss. 

31.  Prove  the  side  of  a  A  is  greater  than  the  difference 
of  the  other  two  sides. 

32.  How  are  the  altitudes  to  the  equal  sides  of  an 
isosceles  triangle  related  ?     Prove  your  answer. 

63. 

Proposition  XXI. 


Given  A  A  B  C  and  P  any  point  within  the  A-  Com- 
pare X  -\'  y  with  a.  Compare  x  ^r  ^  with  b.  Compare  y  ^  z 
with  c. 

Can  you  now  prove  that.^  -V  y  -\-  z  >  J  (  a-f-<5+^)? 

Write  the  general  truth  and  call  it  Prop.  XXI. 


PARALLELS. 


59 


PARALLELS. 

.    64. 

Proposition  XXII. 

How  many  straight  lines  can  be  drawn  through  a  given 
point  parallel  to  a  given  straight  line?  Is  your  answer  self- 
evident? 

[See  §  14,  12.] 

If  two  straight  lines  in  the  same  plane  are  1  to 
to  the  same  line,  can  you  prove  these  two  lines  ||  ? 

If  they  are  not  ||,  can  you  show  that  any  P.  P.  is  violated  ? 
Write  the  formal  statement  of  the  truth  proved  and  call 
it  Prop.  XXII. 

65. 

Proposition  XXIII. 

In  how  many  ways  can  you  draw  a  line  through  a  given 
point  parallel  to  a  given  line? 

Show  them  and  try  to  prove  them. 


A 

m 

P 

n    ,  '  ' 

,  '     '   Q 

H'  '  ' 

8 


Given  m  and  ;?  2  ||  lines,  and  A  B  1  to  »2  at  P. 
Is  A  B  1  to  n} 

If  we  suppose  that  n  is  not  1  to  A  B  at  Q,  draw  H  I  through 
Q  1  to  A  B      How  is  H  I  related  to  mf    But  how  is  n  related 


60  PLANE  GEOMETRY.     BOOK  I. 

to  m?    What  then  is  true  of  the  line  H  I  and  nf     Write  the 
proposition  and  call  it  Prop.  XXIII. 

Exercise. 
33.     Draw  from  any  point  a  1  to  2  ||  lines. 

66. 

Proposition  XXIV. 

E- . F 

A— B 

c D 

Given  E  F  and  C  D  ||  to  A  B. 

Can  you  prove  E  F  and  C  D  j  to  each  other? 

[///;?/.— Draw  a  1  to  A  B.] 

Write  the  general  statement  and  call  it  Prop.  XXIV. 

67. 


Given  any  2  lines,  A  B  and  C  D,  cut  by  a  third  line,  E  F 
How  many  angles  are  formed?  If  there  were  three  lines  cut 
by  E  F,  how  many  Z  s  would  be  formed? 

E  F  is  called  a  transversal  or  secant  line. 

Write  carefully  a  definition  of  a  transversal. 


PARALLELS.  61 

Name  the  vertical  Z  s.  Which  Z  s  lie  between  the  lines 
cut?     These  are  called  interior  Z  s. 

Which  are  the  exterior  Z  s?  Which  are  the  interior  Z  s  on 
the  same  side  of  the  transversal?  Which  are  the  alternate 
interior  Z  s?  Which  are  the  alternate  exterior  Z  s?  Z  2  and 
Z  6  are  called  corresponding  Z  s.  Name  the  other  correspond- 
ing Zs. 

68. 

Proposition  XXV. 


Given  A  B  and  C  D  ||,  cut  by  E  F.  Bisect  that  part  of  the 
line  K  F  included  between  A  B  and  C  D.  From  the  mid-point 
draw  a  1  to  A  B.  Extend  the  line  until  it  reaches  C  D.  How 
is  this  line  related  to  C  D?    How  are  the  2  As  related? 

How  is  Z  3  related  to  Z  6?  How  is  Z  4  related 
to  Z  5  ?  What  kind  of  Z  s  are  3  and  6?  Z  s  4  and  5? 
Can  you  write  a  general  statement? 

Call  it  Prop.  XXV. 

69. 

Cor.  I.  How  are  Z  2  and  Z  6  related  under 
§  68?  What  are  the  other  pairs  of  corresponding  Z  s  ? 
Are  they  equal?     Can  you  state  the  general  truth? 

70. 

Cor,  II.     Compare  Z  1  and  Z  8,  also  Z  2  and  Z  7« 


62 


PLANE  GEOMETRY.      BOOK  I. 


71. 

Cor.  III.     What  is  the  sum  of  Z  5  and  Z  6?    of 
Z4and  Z  6?  of  Z  3  and  Z  6? 

72. 

Proposition  XXVI. 

Problem,      To  draw  through  a  given  point  a  line 
parallel  to  a  given  line.     [Use  §  6  4] 

Exercise. 

34.     Draw  through  the  vertex  of  a  A  a  line  parallel  to 
the  base. 

73. 

Proposition  XXVII. 


Given  2  lines,  A  B  and  C  D,  not  known  to  be  ||,  cut  by  the 
third  line,  E  F,  so  that  Z  4  =  Z  5.  What  other  Z  s  are  equal? 
What  pairs  of  Z  s  =  2  right  Z  s? 

Note  carefully  how  many  deductions  may  be  made  when 
Z-t=  Z5. 

Construct  another  figure  like  the  one  above.  Erect  a  1 
to  A  B  from  the  mid-point  of  that  part  of  the  transversal 
included  between  the  lines  A  B  and  C  D.  Produce  this  1  to  C  D. 
How  are  the  2  As  related? 


PARALLELS.  63 

Can  you  now  show  how  A  B  and  C  D  are  related? 

Write  a  general  statement  of  the  truth  proved  and  call  it 
Prop.  XXVII. 

74. 
Cor.  /.     State  and  prove  the  converse  of  §  69. 

75. 
Cor,  II,     State  and  prove  the  converse  of  §  70. 

76. 

Cor,  III     State  and  prove  the  converse  of  §  71. 

Construct  Prop.  XXVI.,  using  §73,  etc. 
TRIANGLES. 

77. 

Proposition  XXVIII. 


Given  the  A  A  B  C.     Produce  any  side  B  A  forming  the 
exterior  Z  D  A  C.     Draw  w  «  ||  to  B  C. 

Can  you  prove  the  exterior  angle  equal  to  the 
sum  of  the  angles  B  and  C  ? 

Does  this  prove  a  general  proposition?     State  it.     Call  it 
Prop.  XXVIII. 

[Let  the  pupil  prove  when  A  B  is  produced.] 


64  PLANE  GEOMETRY.     BOOK  I. 

78. 

Proposition  XXIX. 

What  IS  the  value  of  the  sum  of  the  Z  s  of  any  A? 
Prove  your  answer.     See  figure  in  Prop.  XXVIII. 
Write  Prop.  XXIX. 

EXERCISEJS. 

35.  Why  can  a  A  not  have  2  right  Z  s?  What  13  the 
sum  of  the  2  acute  Z  s  of  a  right  A?  What  is  the  relation 
of  the  two  acute  Z  s  of  a  right  A? 

36.  If  2  Z  s  of  a  A  are  equal  respectively  to  2  Z  s  of  an- 
other A,  how  do  the  third  Z  s  compare? 

37.  If  in  2  right  As  an  acute  Z  of  one  equals  an  acute  Z 
of  the  other,  how  are  the  other  acute  Z  s  related? 

79. 

Proposition  XXX. 

Quote  the  propositions  that  prove  2  right  As  equal  to 
each  other. 

Construct  a  right  scalene  A  A  B  C,  having  the  right  Z  at 
B.  Construct  another  right  A  A'  B'  C,  having  the  right  Z  at 
B'  and  the  side  A'  B'  equal  to  side  A  B  and  Z  at  A'  =  Z  at  C. 

Can  you  prove  the  triangles  equal? 

Is  it  possible  to  draw  a  right  A  A'  B'  C  equal  to 
the  right  A  A  B  C  when  only  one  side  and  one  acute 
Z  of  the  first  equals  a  side  and  an  homologous  acute 
Z  of  the  second? 

Can  you  discover  a  new  proposition  about  the  equality  of 
2  right  As? 

Write  a  clear  statement  of  the  truth  discovered  and  calliit 
Prop.  XXX. 


TRIANGLES.  65 

80. 

Proposition  XXXI. 

Given  the  A  A  B  C  with  Z  A  =  Z  B. 

Is  the  A  isosceles? 

{Hint. — Drop  a  1  from  C  to  the  base  A  B.] 
Write  Prop.  XXXI. 

Exercise 
38.     Prove  that  an  equiangular  A  is  equilateral. 

81. 

Proposition  XXXIT. 


Given  A  A  B  C,  in  which  Z  B  >  Z  A. 

Compare  the  sides  opposite  the  unequal  Z  s.  Are 
they  equal.     If  not,  which  is  greater? 

Prove  your  answer. 

[Ffint. — At  B  and  abgve  A  B  construct  an  Z  equal  to  Z  A 
having  A  B  for  one  side.  Must  the  side  fall  within  B  C?  Why? 
Letter  point  of  intersection  with  A  C,  D.  Does  A  D  =  B  D? 
Does  ADC  =-B  DC?  Is  B  D  C  >  B  C?  Give  reason  for 
each  step.] 


66  PLANE  GEOMETRY.      BOOK  I. 

82. 
Proposition  XXXIII. 
Given  A  A  B  C,  in  which  A  C  >  B  C. 
Compare  the  Z  s  opposite  the  unequal  sides. 

(1)  Either  Z  B  =  Z  A, 

(2)  or  Z  B  <   Z  A, 

(3)  or  Z  B  >   Z  A. 

Show  that  (1)  and  (2)  are  impossible;  hence  (8)  must  be 
true. 

This  proof  is  based  on  the  doctrine  of  exclusion. 

83. 

Proposition  XXXIV. 

Draw  two  As,  ABC  and  D  E  F,  making  sides  A  B  and 
A  C  respectively  equal  to  sides  D  E  and  D  F,  but  the  Z 
at  A  greater  than  the  Z  at  D.  Now  place  A  D  E  F  upon  A 
A  B  C  so  that  D  falls  upon  A  and  D  E  takes  the  direction  of 
AB.  Where  must  E  fall?  Why?  Where  must  D  F  fall?  Why? 
Does  the  third  side  of  the  second  A  at>pear  to  be  as  long  as  the 
third  side  of  the  first  A?  Our  problem  requires  us  to  prove 
which  is  the  longer.  If  we  bisect  Z  C  A  F  and  letter  the  new 
point  on  B  C,  G,  and  join  F  to  G,  we  have  two  new  As,  G  A  C 
and  G  A  F.  Are  they  equal?  Why?  Does  G  F  =  G  C? 
Why? 

Can  you  now  prove  which  is  greater,  B  C  or  E  F? 

Write  Prop.  XXXIV. 

84. 
Proposition  XXXV. 

What  is  the  converse  of  Prop.  XXXIV.? 

Try  to  prove  this  by  supposing— (1)  that  the  included  Z  s 
are  equal;  (2)  that  the  included  Z  in  the  first  A  is  less  than 
the  included  Z  in  the  second  A-  Write  Prop.  XXXV.  in 
the  notes. 


PARALLELOGRAMS.  67 


PARALLELOGRAMS. 

What  is  a  quadrilateral? 

What  are  its  diagonals?  What  is  meant  by  the  angles  of  a 
quadrilateral? 

Make  a  quadrilateral  the  angles  of  which  shall  each  be 
less  than  a  straight  angle. 

Can  you  make  a  quadrilateral  having  an  angle  greater  than 
a  straight  angle? 

How  does  this  figure  diflfer  from  the  first? 

What  name  is  given  to  a  quadrilateral  whose  opposite 
sides  are  parallel?  if  only  two  sides  are  parallel  ?  if  no  two 
sides  are  parallel? 

What  are  the  bases  of  a  parallelogram?  of  a  trapezoid? 

Note. — Remember  that  all  rectangles  are  parallelograms, 
but  all  parallelograms  are  not  rectangles.  A  rectangle  should 
not  be  used  in  proving  properties  about  parallelograms.   Why? 

85. 

A  diagonal  of  a  polygon  is  a  line  joining  the  vertics  of 
two  angles  not  adjacent. 

86. 

A  trapezium  is  a  quadrilateral  having  no  sides  parallel. 

The  "kite"  trapezium  has  two  pairs  of  equal  sides  and 
each  angle  less  than  a  straight  angle.  The  "arrow"  trapezium 
has  two  pairs  of  equal  sides  and  one  of  its  angles  is  reflex  or 
greater  than  a  straight  angle.  Draw  trapeziums,  illustrating 
each  class. 

87. 

A  trapezoid  is  a  quadrilateral  having  only  two  sides 
parallel. 

1.  The  non  parallel  sides  are  called  the  legs  of  the 
trapezoid. 


68-  PLANE   GEOMETRY.      BOOK  I. 

2.  The  parallel  sides  are  called  the  bases. 

3.  Draw  a  trapezoid  in  which  the  legs  are  equal.  It  is 
called  an  isosceles  trapezoid.  Draw  a  trapezoid  containing  a 
rt.  Z. 

88. 

A  rhomboid  is  a  parallelogram  having  adjacent  sides  un- 
equal and  angles  oblique. 

89. 

A  rhombus  is  a  parallelogranj  having  adjacent  sides  equal 
and  angles  oblique. 

90. 

Definition. — An  i?itercepi  is  a  line  or  a  part  of  a  line  inter- 
cepted between  two  other  lines. 

91. 

Proposition  XXXVI. 

Review  the  definition  of  a  parallelogram.  Remember 
that  in  any  figure  which  is  given  a  parallelogram  you  must 
assume  no  relations  of  the  sides  and  angles  which  are  not 
warranted  in  the  definition. 


Given:     The  parallelogram  A  B  C  D  and  its  diagonal  A  C. 

Are  the  triangles  into  which  the  parallelogram 
is  divided  by  the  diagonal,  equal .^ 

[Prove  when  diagonal  B  D  is  drawn.] 


PARALLELOGRAMS.  69 

92. 

Cor.  I.     Prove  that  the  opposite  sides  of  a  a  are 
equal. 

93. 

Cor.  II.     Prove  that  the  opposite  angles  of  a  ci 
are  equal. 

Question:     State  the  converse  of  Prop.  XXXVI.     Is  it 
always  true? 

94.       ' 

Cor,  III,     Prove  that  parallel  intercepts  between 
parallel  lines  are  equal. 

Exercises. 

39.  Give7i:     Parallelogram  A  B  C  D,  in  which   Z  B  is  a 
rt.  Z  .     Make  deduction  and  prove. 

40.  Given:     Parallelogram  A  B  C  D,  in  which  Z  D  is  a 
rt.  Z  and  side  A  B  =  side  B  C.     Make  deduction  and  prove  it 

41.  Draw  the  bisectors  of  two  opposite  angles  of  a  paral- 
lelogram.    Prove  that  these  bisectors  are  parallel. 

(If  you  fail,  see  "hint"  below.) 

\^Hi7it. — Produce  bisectors  until  sides  of  the  parallelogram 
are  cut  and  prove  the  new  figure  a  parallelogram.] 


70  PLANE  GEOMETRY.     BOOK  I. 

95. 

Proposition  XXXVII. 
Draw  a  parallelogram  and  both  diagonals. 
Compare  the  parts  of  each  diagonal. 
Make  deduction.     State  and  prove  Prop.  XXXVII. 

ExERCivSES. 

42.     Given   the   rhomboid  A  B  C  D  and   the   diagonals 
A  B  and  B  D  intersecting  at  E ;  also  A  B  >  B  C. 

(1)  Compare  the  four   /\s.     State  and  prove  which 
are  equal. 

(2)  Compare  Z  s  C  K  B  and  B  H  A.     Prove  deduc- 
tions. 

(3)  Why  musl  the  diagonals  of  a  rhomboid  be  un- 
equal? 

48.     Draw  a  rhombus  and  both  diagonals. 

(1)  Compare  the  Z  s  at  intersection. 

(2)  Compare  the  four  triangles. 

(3)  Why  must  the  diagonals  be  unequal? 
Prove  all  deductions  made. 

44.  Draw  a  rectangle  and  both  diagonals. 
.   (1)     Compare  the  diagonals. 

(2)  Compare  the  four  triangles  in  order. 

(3)  Make  a  deduction  which  is  true  of  each  of  the 
four  triangles. 

(4)  Compare  the  rectangle  with  the  rhomboid  and 
state  points  of  difference  and  similarity. 

45.  Draw  a  square  and  its  diagonals. 

(1)  Make  and  prove  deductions. 

(2)  Make  careful  comparison  of  square  and  rhombus. 


PARALLELOGRAMS.  71 

96. 

Proposition  XXXVIII. 


Given:  1.  The  quadrilateral  A  B  C  D  and  the  diagonal 
A  C,  forming  l_^  x,  m,  7i  and  y. 

2.     Also  (I)  A  B  =  D  C,  and  (2)  A  D  =  B  C. 

Is  the  quadrilateral  a  parallelogram? 

State  and  prove  Prop.  XXXVIII. 

(If  you  fail,  see  "hint"  below.) 

[Hint. — Prove  As  equal;  consequently  Lm  =  /_n  and 
A  B  parallel  to  D  C.  Then  prove  B  C  parallel  to  A  D,  and 
tell  why  the  figure  is  a  parallelogram.] 


97. 

Proposition  XXXIX. 

Given  the  quadrilateral  A  B  C  D,  in  which  A  D  =  B  C 
and  A  D  is  parallel  to  B  C. 

Can  you  prove  the  figure  a  parallelogram? 

(If  you  fail,  see  **hint"  below.) 

[Hi7it. — Draw  either  diagonal  and  use  method  given  in  the 
''hint"  to  §96.] 


72  PLANE  GEOMETRY.      BOOK  I. 

98. 

.  Proposition  XL. 

Draw  two  parallelograms  A  B  C  D  and  A'  B'  C  D',  in 
which  any  two  adjacent  sides  and  the  included  angle  of  the 
one  equal  respectively  two  adjacent  sides  and  the  included 
angle  of  the  other. 

Are  the  parallelograms  equal? 

[N.  B. — Equal  figures  can  be  made  to  coincide  in  all  parts. 
Figures  may  be  equal  in  area,  or  equivalent,  which  are  not 
equal.^ 

State  and  prove  Prop.  XL. 

(If  you  fail,  consult  ''hint"  below.) 

[yy2«/.— Superpose  A'  B'  C  D'  on  A  B  C  D  so  that  the 
equal  sides  and  the  included  Z  s  will  coincide.  Then  show 
that  the  opposite  sides  and  the  remaining  vertex  must  coin- 
cide.    See  §  14,  12,  and  §  14,  10,] 

Exercises. 

46.  If  the  diagonals  of  a  quadrilateral  bisect  each  other, 
what  is  the  figure  ?  Distinguish  in  the  above  when  the  diag- 
onals are  (1)  equal,  (2)  unequal. 

47.  If  the  diagonals  of  a  quadrilateral  are  (1)  equal, 
(2)  unequal,  and  bisect  each  other  at  right  angles,  what  deduc- 
tions can  you  make? 


Questions. 

1.  If  the  diagonals  of  a  quadrilateral  are  equal,  can  you 
make  and  prove  any  deduction? 

2.  If  the  opposite  angles  of  a  quadrilateral  are  equal, 
what  deduction  can  you  make  and  prove? 


PARALLELOGRAMS.  73 

Exercises. 

48.  By  drawing  the  diagonals  A  C  and  A'  C  in  §  98  can 
you  prove  Prop.  XL.  in  another  way? 

49.  Given  the  A  A  B  C,  in  which  /  A  plus  twice  Z  B 
minus  three  times  Z  C  equals  110®,  and  Z  A  minus  twice 
Z  B  plus  Z  C  equals  90°;  find  Z  s  A,  B,  C. 

50.  Given  two  rectangles  having  the  base  and  altitude  of 
one  respectively  equal  to  the  base  and  altitude  of  the  other; 
show  how  they  compare  in  area. 

51.  What  is  the  value  of  the  sum  of  all  the  angles  of  a 
parallelogram? 

99. 

Proposition  XLI. 

Problem,  To  construct  a  parallelogram  when 
two  adjacent  sides  and  the  included  angle  are  given. 

Exercises. 

52.  Construct  a  rectangle  when  a  side  is  given  and  the 
adjacent  side  is  2^  times  the  given  side. 

53.  Construct  a  square  when  a  diagonal  is  given.  [Show 
two  ways.] 

54.  Construct  a  rhombus  when  the  diagonals  are  given. 

55.  Construct  a  rhombus  when  one  diagonal  is  given  and 
the  other  is  3^  times  ito 

56.  Construct  a  "kite"  trapezium  when  the  diagonals 
arc  given.  Discuss  the  possible  lengths  and  position  of  the 
diagonals. 

57.  Construct  an  isoscetes  trapezoid  when  a  diagonal 
and  distance  between  the  bases  are  given.    Discuss, 


V4  PLANE   GEOMETRY.      BOOK  I. 

POI.YGONS. 

What  is  a  polygon?  its  perimeter?  Compare  the  "arrow" 
trapezium  with  the  "kite"  trapezium.  What  is  a  convex  pol- 
ygon? a  re-entrant  angled  or  concave  polygon?  What  is  a 
regular  polygon?  the  angle  of  a  regular  polygon?  the  an- 
gle at  the  center?     What  is  an  exterior  angle  of  a  polygon? 

What  are  the  diagonals  of  a  polygon?  How  are  pol3^gons 
classified? 

100. 

Polygons  are  classified  according  to  the  number  of  sides. 
A  polygon  of  three  sides  is  a  triangle^  or  trigon;  of  four  sides  is 
a  quadrilateral;  of  five  sides  is  s.pe7itagon;  of  six  sides  is  a  hexa- 
gon; of  seven  sides  is  a  heptagon;  of  eight  sides  is  an  octagon; 
of  nine  sides  is  a  nonagon;  of  ten  sides  is  a  decagon;  of  eleven 
sides  is  an  undecagon;  of  twelve  sides  is  a  dodecagon. 

101. 

The  sum  of  the  sides  of  a  polygon  is  called  tho:  perimeter. 

102. 

A  C071V ex  polygon  is  one  in  which  no  side  produced  will 
enter  the  polygon. 

103. 

A  concave  polygon  is  one  in  which  at  least  two  sides  when 
produced  enter  the  polygon.  The  angle  whose  sides  produced 
enter  the  polygon  is  called  a  re-entrant  angle. 

104. 

A  regular  polygon  is  both  equilateral  and  equiangular. 
When  the  term  polygon  is  used,  a  convex  polygon  is  meant- 


POLYGONS.  75 

105. 

Proposition  XLH. 

Into  how  many  triangles  may  we  divide  any  polygon  by 
joining  any  vertex  to  all  other  vertices  when  there  are  four 
sides?  five  sides?  six  sides?  7i  sides? 

What  is  the  sum  of  all  the  interior  angles  of  a  polygon  of 
four  sides?  five  sides?  six  sides?  71  sides? 

The  sum  of  the  interior  angles  of  a  polygon  is 
equal  to  as  many  straight  angles  as  — 

Finish  the  above  statement  and  prove  it.  Call  it  Prop. 
XLII. 

106. 

Cor,  I.  If  the  sides  of  a  polygon  are  produced 
in  order,  the  exterior  angles  thus  formed  are  equal 
to  two  straight  angles. 

[Hint. — What  is  the  sum  of  the  interior  angles?  of  the 
interior  and  exterior  angles  taken  together  ?] 


107. 

Cor.  II,     If  a  polygon  is  equiangular,  each  in- 
terior angle  is  equal  to  as  many  straight  angles  — 

Finish  and  prove. 

[Hi7it. —  What  is  the  sum  of  the  Zs  of  any  triangle?  Then 
what  is  the  size  of  each  angle  of  an  equiangular  triangle? 
What  is  each  angle  of  an  equiangular  quadrilateral?  etc.] 

Write  the  fraction  which  represents  the  num- 
ber of  St.  Zs  in  each  Z  of  a  polygon  of  n  sides. 


76 


PLANE  GEOMETRY.     BOOK  I. 


108. 


Proposition  XLIII. 

I^et  C  E  be  the  right  bisector  to  the  line  A  B;  i.  e.,al  at  the 
mid-point  of  A  B.  Take  any  point  P  in  the  right  bisector 
and  join  it  with  the  extremities  of  A  B. 

(1)  Compare  P  A  and  P  B. 

Take  any  point  P'  outside  the  right  bisector  and  join  it 
with  the  extremities  of  the  line  A  B. 

(2)  What  can  you  prove  concerning  P'  A  and 
P^B? 


(If  unable  to  prove  the  second  part  of  the  above,  see 
"hint"  below.) 

[f^inL — Let  P'  be  without  and  to  the  right  of  the  right 
bisector  C  K. 

Draw  P'  B  and  F  A; 

To  Prove  F  B  <  P'  A. 

Since  P'  is  to  the  right  of  the  right  bisector,  P'  A  will 
intersect  it;  letter  the  point  of  intersection  M. 

Draw  M  B. 

M  A  =  M  B,     [?] 

And  M  B  +  M  P'  >  P'  B,     [?] 

And  M  A  +  M  P'  >  P'  B,     [?] 

Or  A  P'  >  P'  B.     \J] 


BISECTORS.  77 

Let  the  pupil  take  P'  to  the  left  of  the  right  bisector  and 
prove  P'  B  and  P'  A  are  unequal. 

109. 

Distance  from  a  point  to  a  line  is  measured  by  a  perpen- 
dicular drawn  from  the  point  to  the  line.  The  distance  be- 
tween parallel  lines  is  a  perpendicular  from  any  point  in  one 
line  to  the  other  line. 

110. 

Proposition  XLIV. 

I.  Given  any  angle  A  C  B  and  its  bisector  D  C;  also  P 
any  point  in  the  bisector. 

Compare  the  distances  from  P  to  the  sides  of  the 
sides  of  the  angle,  C  B  and  C  A 

[Hint. — How  do  you  measure  distance  from  a  point  to  a 
straight  line?  Draw  the  required  lines  and  compare  the  As 
formed.] 

II.  Given  any  angle  A  C  B  and  its  bisector  D  C;  also  P 
any  point  without  the  bisector. 

Compare  the  distances  from  P  to  the  sides  of  the 
angle,  C  A  and  C  B. 


Let  P  be  below  the  bisector  C  D.     Draw  Is,  P  Q  and  P  R, 
to  the  sides  of  the  angle,  C  B  and  C  A  respectively. 
Then  prove  P  Q  <  P  R. 


78  PLANE  GEOMETRY.      BOOK  I. 

Since  P  is  below  C  D,  P  R  will  intersect  C  D;  letter 
point  of  intersection  E. 

Draw  K  M  1  to  C  B; 

K  R  =  K  M.     [?J 

Also  draw  MP; 

:  M  P  >  P  Q.     [?] 

But  E  M  4-  E  P  >  M  P;     [?] 

.-.  E  M  H-  EP>PQ,     [?] 

And  E  R  +  E  P  >  P  Q,     [?] 

Or  P  R  >  P  Q.     [?] 

L^t  the  pupil  take  any  point  above  the  bisector  C  D  and 
prave  that  the  distances  to  the  sides  of  the  Z  are  unequal. 

Write  Prop.  XlylV.,  the  proof  of  which  includes  both  of 
the  above  proofs. 

111. 

State  and  prove  the  converse  of  Prop.  XLIII. 

112. 
State  and  prove  the  converse  of  Prop.  XLIV. 
113. 
Proposition  XLV. 
Theorem.     The  three  right  bisectors  of  the  sides 
of  any  triangle  meet  in  a  common   point,  which  is 
equidistant  from  the  vertices  of  the  triangle. 

114. 

Proposition  XLVI. 
Theorem.     The  three  bisectors  of  the  angles  of 
any  triangle  intersect  in  a  common  point  which  is 
equidistant  from  the  three  sides  of  the  triangle. 

115. 

When  three  or  more  lines  intersect  in  a  common  point 
they  are  said  to  be  concurrent. 


TRANSVERSALS. 

TRANSVERSAI.S. 

116. 

Proposition  XLVII. 


79 


Let  A  B,  C  D,  etc.,  be  ||  lines  cut  by  the  transversal  ;i;jkso 
that  the  parts  intercepted,  I  J,  J  K,  etc.,  are  equal.  Now  draw 
any  other  transversal  m  7i  Do  the  parts  intercepted  by  the 
||s  on  w  w  appear  to  be  equal? 

Through  points  I,  J,  etc.,  draw  lines  ||  to  m  n.  Does  I  S 
=  O  P?  Why?  Make  further  deductions.  Can  you  prove 
IS=JT  =KU? 

[Rint.—Kx^  As  I  S  J,  J  T  K,  etc  ,  equal?     Why?] 

What  then  can  you  prove  about  parts  intercepted 
on  m  nf     Do  you  discover  any  general  truth  .^ 

Carefully  retrace  the  steps  by  which  yoti  reach  the  con- 
clusion and  write  Prop.  XLVII. 

117. 

Proposition  XLVII  I. 

Problem.      To  divide  a  line  into  n  equal  parts. 

(See  "hint"  if  necessary.) 

\Hint.—\^^\.  71  equal  5  or  some  known  number.  Draw  an 
indefinite  line  making  an  angle  with  the  given  line  at  either 
end.     Lay  off,  beginning  at  the  vertex,  5  equal  parts  on  the 


80  PLANE  GEOMETRY.      BOOK   I. 

indefinite  line  and  join  the  remote  end  of  the  last  part  with  the 
other  end  of  the  given  line.  Parallel  to  this  line  draw  through 
the  points  of  division  other  lines  cutting  the  given  line.  Prove 
that  the  parts  intercepted  by  the  parallels  are  the  required 
equal  parts.] 


118. 

Proposition  XLIX. 


Suppose  ABC  any  /\.  Through  m,  the  middle  point  of 
any  side,  A  B,  draw  a  line  ||  to  the  side  B  C,  cutting  A  C  at 
n.  Through  A  draw  ;i;  j/  ||  to  B  C.  How  many  ||  lines  have 
you?     Why? 

(1)  Does  A  n  appear  to  be  equal  to  n  C?  Can 
you  prove  your  supposition? 

(2)  What  part  of  B  C  does  m  n  appear  to  equal? 

Do  you  see  any  way  to  prove  your  answer?  State  Prop. 
XLIX.     There  are  two  parts. 

(See  "hint"  below  if  you  fail) 

Let  the  pupil  use  either  of  the  other  sides  for  the  base 
and  prove  the  above  proposition. 

[Hint. — Draw  through  n  a  line  parallel  to  A  B.] 


POLYGONS.  81 

119. 

Proposition  L. 

(1)  If  the  middle  points  of  two  sides  of  a  trian- 
gle are  joined,  will  the  line  joining  these  points  be 
parallel  to  the  third  side? 

[Hint. — Through  one  of  the  mid-points  draw  a  line  ||  to 
the  third  side.     Prove  the  lines  must  coincide.] 

(2)  What  part  of  the  base  does  the  line  joining 
the  mid-points  of  the  sides  of  a  triangle  equal? 

Write  the  two  truths  as  Prop.  L. 

120. 

The  line  joining  the  mid-points  of  the  legs  of  a  trapezoid 
is  called  the  median  of  the  trapezoid. 

121. 

Draw  any  trapezoid,  and  through  the  mid-point  of  either 
leg  draw  a  line  parallel  to  either  base. 

Cor.  /.  (1)  How  does  this  parallel  meet  the 
other  leg?  (2)  Compare  the  length  of  the  parallel 
to  the  sum  of  the  bases. 

Write  the  two  truths  discovered  as  Cor.  I.  to  Prop.  L. 

122. 

Cor.  II.    Draw  any  trapezoid  and  its  median. 

What  relations  does  the  median  bear  to  the  two 
bases  .^ 

Prove  your  answer.  Write  the  two  truths  discovered  as 
Cor.  II.,  Prop.  L. 


PLANE  GEOMETRY.     BOOK  I. 

123. 

Proposition  lyl. 


WM 


I.  Given  the  acute  A  A  B  C  with  the  altitudes  B  R,  C  S, 
A  T.  Through  the  vertices  C,  A,  B  draw  lines  A'  B',  B'  C\ 
C  A',  respectively,  ||  to  the  sides  A  B,  B  C,  C  A. 

In  the  figure  A  B  C  B',  how  are  A  B'  and  B  C  related? 
Why?     How  are  A  B  and  B'  C?     Why? 

What  figure  is  A  B  C  B?  Prove  in  similar  manner  A  C 
=  B  C.     How  are  A  B'  and  A  C  related  ?     [Ax.] 

Prove  C  B  and  B  A'  each  equal  to  A  C.  Compare  B  C' 
and  B  A'. 

Prove  B'  C  and  A'  C  equal  to  each  other. 

What  are  A  T,  C  S,  and  B  R  to  the  A  A'  B'  C? 

By  P.  P.  what  have  we  proven  concerning  these  three 
lilies?    But  what  are  these  three  lines  of  the  A  A  B  C? 

What  then  can  you  prove  of  the  altitudes  of  any 
acute  triangle? 


MEDIANS. 


II.     CotiStruct  an  oblique  A  with  two  of  the  altitudes  fall- 
ing without  the  A-     Prove  that  the  altitudes  are  concurrent. 

Exercise. 
58.     The  Is  and  the  Is  produced  from  two  opposite  ver- 
tices of  a  rhomboid  to  the  sides  opposite,  or  to  those  sides 
produced,  form  with  the  sides  of  the  rhomboid  and  the  sides 
produced,  two  rectangles  and  two  rhomboids. 


124. 

Proposition  LII. 


Given  the  oblique  A  A  B  C  and  the  medians  AD,  B  F, 
and  C  H. 

I^et  P  be  the  point  of  intersection  of  any  two  medians, 
C  E  and  A  D,  and  take  H  the  middle  of  A  P,  and  G  the  mid- 
dle of  C  P.     Draw  H  G,  G  D,  D  E,  E  H. 


84  PLANE  GEOMETRY.     BOOK  I. 

In  the  A  A  B  C,  how  is  B  D  related  to  A  C?  How  does 
E  D  compare  in  length  with  A  C?  Why?  In  A  A  P  C,  how 
does  H  G  compare  with  A  C?  Compare  H  G  and  E  D.  What 
is  the  figure  H  E  D  G?  Why?  Compare  H  P,  P  D,  and  G  P, 
P  E.  Compare  D  P  with  A  D  and  E  P  with  E  C.  How  do 
the  two  medians  A  D  and  C  E  cut  each  other? 

Since  A  D  and  C  E  are  any  two  medians,  how  will  B  F 
cut  C  E  or  A  D? 

What  two  truths  have  we  discovered  about  the 
three  medians  of  any  triangle? 

State  these  truths  formally  and  number  the  Prop.  LII. 

EXKRCISES. 

59.  Join  the  middle  points  of  the  sides  of  a  A-  What 
new  "figures  do  you  get?  Prove  how  they  are  related  to  the 
original  A- 

60.  How  does  the  median  drawn  to  the  hypotenuse  of  a 
rt.  A  compare  wuth  the  hypotenuse? 

[Hint. — Draw  the  rt.  A  and  the  required  median.  Draw 
through  mid-point  of  the  hypotenuse  a  line  parallel  to  the 
base.  How  will  this  parallel  meet  the  altitude  of  the  rt.  A  ? 
Compare  the  parts  of  the  altitude.  Do  you  not  now  see  the 
relation  of  the  median  to  the  hypotenuse?] 

61.  One  angle  of  an  isosceles  A  is  60°.  What  can  you 
prove  about  the  A? 


LOCI. 


LOCI. 

What  have  you  learned  about  every  point  in  the  line 
bisecting  a  given  Z  ?  How  then  could  you  find  a  line  which 
contains  all  the  points  equidistant  from  the  sides  of  a  given  Z  ? 

Can  you  find  a  ball  when  it  is  known  to  be  on  the  ground 
just  20  yards  from  the  foot  of  a  given  tree?  Can  you  find  a 
line  which  contains  all  the  points  which  are  a  given  distance 
from  a  given  point? 

Can  you  find  two  lines  which  contain  all  thq  points  which 
are  a  given  distance  from  a  given  line? 

Can  you  find  a  line  which  contains  all  the  points  equidis- 
tant from  the  extremities  of  a  given  line  ? 

126. 

Definition. — The  place,  line,  or  system  of  lines  which  con- 
tains all  the  points,  and  only  those  points,  which  satisfy  a 
given  condition,  is  called  the  loc2is  of  the  point. 

Exercises. 

What  is  the  locus  of  the  point: 

62.  Equally  distant  from  two  given  points? 

63.  Equally  distant  from  two  given  straight  lines — 
{a)    When  the  lines  intersect  ?    (b)  When  the  lines  are  parallel  ? 

64.  Equally  distant  from  the  extremities  of  a  given  st. 
line? 

65.  Equidistant  from  three  given  points,  A,  B,  C,  which 
are  not  in  the  same  st.  line? 

66.  A,  B,  C  are  three  towns  on  a  st.  road.  B  is  two  miles 
north  of  A  and  four  miles  south  of  C.    Find  a  point  equidis- 


86  PLANE  GEOMETRY.     BOOK  I. 

tant  from  A  and  C  and  four  miles  from  B.     Is  there  only  one? 

67.  What  is  the  locus  of  the  vertex  of  an  isosceles  /\ 
having  a  given  base  ? 

Bach  locus  r^(\mr^S2i  geometrical  j)roo/;  e.  g.,  in  Ex.  62— to 
prove  the  answer,  "The  locus  of  a  point  equidistant  from  two 
given  points  is  the  right  bisector  of  a  line  joining  the  given 
points,"  fix  the  points,  draw  the  line  joining  them,  and  the 
right  bisector.  Then  select  any  point  in  the  locus  (the  right 
bisector)  and  prove  that  it  is  equidistant  from  the  given 
points.  Then  select  any  point  without  the  locus  and  prove 
that  it  is  not  equidistant  from  the  points. 

In  general  we  may  say  that  when  we  prove  a  theorem 
concerning  the  locus  of  points,  it  is  necessary  to  prove  two 
things: 

(1)  That  all  points  in  the  locus  satisfy  the  given  con- 
ditions. 

(2)  That  any  point  not  in  the  locus  does  not  satisfy  the 
given  conditions. 

Can  you  tell  why  both  of  the  above  proofs  are  necessary? 


CIRCLES.  87 


BOOK  II. 

circi.es. 

Define  (I)  a  circle,  (2)  a  circumference,  (3)  an  arc,  (4)  a 
chord,  (5)  a  radius,  (0)  a  diameter. 

Are  all  radii  and  all  diameters  of  the  same  or  equal  circles 
equal? 

127. 

A  segment  of  a  circle  is  a  part  cut  off  by  a  chord.  It  is 
bounded  by  an  arc  and  the  chord  cutting  it  off. 

128. 

A  semi-cirele  is  one-half  of  a  circle.  A  semi-circumference 
is  one-half  of  the  circumference.  A  semi-circle  is  bounded  by 
a  diameter  and  a  semi-circumference. 

129. 

A  sector  is  a  part  of  a  circle  bounded  by  an  arc  and  two  radii. 

130. 

A  tangent  to  a  circle  is  a  line  which  touches  it,  but  will  not 
enter  the  circle,  no  matter  how  far  the  tangent  is  produced. 

The  poifii  of  tangency  is  the  point  where  the  line  touches 
the  circle. 

Two  circles  are  tangent  when  their  circumferences  touch 
but  do  not  cut  each  other.     They  may  be  tangent  irtternally 


88  PLANE  GEOMETRY.      BOOK  II. 

when  one  is  wholly  within  the  other;   or  tangent  externally 
when  one  is  wholly  without  the  other. 

131. 

A  secant  is  a  line  which  cuts  the  circumference  in  two 
points. 

132. 

A  central  angle  is  an  angle  formed  by  two  radii. 

133. 

(1)  An  inscribed  angle  is  an  angle  whose  vertex  is  a  point 
in  the  circumference,  and  whose  sides  are  chords. 

(2)  An  inscribed  polygon  has  its  vertices  in  the  circum- 
ference of  the  circle. 

134. 

Proposition  I. 

Which  chord  divides   the  circle  into  two  equal 
parts? 

Prove  your  answer.     State  Prop.  I. 

135. 

Proposition  II. 

Draw  a  circle  and  any  chord  not  the  diameter. 
Draw  a  radius  perpendicular  to  the  chord. 

Compare  the  parts  of  the  chord. 

State  arid  prove  Prop.  II. 


CIRCLES.  89 

136. 

Cor.  I.  Draw  a  circle  and  a  chord.  Draw  the  right 
bisector  of  th    chord. 

Will  it  pass  through  the  center  of  the  circle  ? 

State  and  prove  Cor.  I.,  Prop  II. 

{^Flint. — If  you  fail  to  prove  the  above,  consult  §  HI.] 

Exercise. 

68.  Construct  a  chord  when  the  circle  and  the  mid-point 
of  the  chord  are  given. 

(See  *'hint"  below  if  you  fail) 

[Hint. — Draw  the  radius  through  the  given  point;  then 
draw  a  1  to  the  radius  at  that  point.] 

137. 

Proposition  III. 

(1)  Draw  in  equal  circles  two  equal  central  angles. 

Do  the  arcs  which  subtend  the  equal  angles 
appear  to  be  equal? 

[How  did  you  prove  that  circles  having  equal  radii  or 
equal  diameters  are  equal?  How  then  can  you  prove  arcs 
equal?] 

{Stibtend  means  to  be  below,  or  under,  or  to  be  opposite;  e.g., 
arcs  are  said  to  subtend  central  angles,  and  chords  subtend  arcs) 

(2)  Again,  in  equal  circles  take  equal  arcs. 

Do  the  central  angles  appear  equal  .^ 

State  the  truths  discovered  as  one  proposition,  the  second 
part  being  the  converse  of  the  first.     Prove  Prop.  III. 


00  PLANE  GEOMETRY.      BOOK  11. 

138. 

Cor.  I.     Draw  a  circle  and  a  chord.     Draw  the  right  bi- 
sector of  the  chord. 

Does  it  pass  through  the  center?     [Auth.] 
Does  it  bisect  the  central  angle? 

Prove  your  answer. 

Does  it  bisect  the  subtended  arc? 

Prove  it. 

State  the  truths  discovered  as  one  corollary. 

KxKRCiSE. 

69.     (1)  Bisect  a  given  arc.     (2)  Bisect  a  given  angle,  not 
as  in  §  46. 


139. 


Proposition  IV. 

Given  two  equal  circles  and  two  equal  arcs. 
Compare  the  chords  which  subtend  the  arcs. 
Again,  given  two  equal  circles  and  two  equal  chords. 
Compare  the  arcs  subtended. 
State  and  prove  Prop.  IV. 

Exercise. 

70.     Construct  and  prove  §49  again,  using  a  method  you 
were  unable  to  use  when  first  given. 


CIRCLES.  91 

140. 
Proposition  V. 

Problem.      To  find  the  center  of  a  given  circle. 
[i^/?i/.— See§135.] 

Exercise. 

71.     Find  the  center  of  a  circle  when  only  an  arc  is  given. 
[Can  you  find  the  ccijter  when  only  a  chord  is  given?] 

141. 

Proposition  VI. 

Problem.      To  construct  an  arc  equal  to  a  given 
arc. 

142. 

Proposition  VII. 

How  do  you  find  the  distance  from  a  chord  to  the  center 
of  the  circle? 

Draw  two  equal  circles  and  two  equal  chords. 

Compare  the  distances  of  these  chords  to  the 
centers  of  the  circles. 

State  and  prove  the  converse  of  the  above. 
Write  both  conclusions  as  Prop.  VII. 

143. 

Proposition  VIII. 

Given  in  equal  circles  unequal  arcs,  each  being  less  than 
a  semi-circumference. 

Compare  the  central  angles. 

State  and  prove  the  converse. 


92  PLANE  GEOMETRY.      BOOK  II. 

144. 

Proposition  IX. 

Given  in  equal  circles  unequal  arcs,  each  being  less  than 
a  semi-circumference. 

Compare  the  chords  subtending  the  arcs. 

[Which  has  the  greater  central  angle  ?  Compare  the 
triangles  formed.] 

State  and  prove  the  converse, 
[Use  doctrine  of  exclusion.] 

145. 
Proposition  X. 

How  many  circumferences  can  be  made  to  pass 
through  any  three  points  which  are  not  in  the  same 
straight  line  ? 

[Hint. — To  discover  a  solution,  suppose  the.  problem 
solved.  Draw  a  O  and  consider  any  three  points  on  the 
circumference  the  given  points.  Study  the  figure.  How 
could  you  replace  the  circumference  were  it  erased  ?  Could 
you  find  the  center  of  the  O  ?  What  proposition,  corollary, 
or  exercise  declares  that  certain  lines  pass  through  the 
center?  Have  you  the  data  necessary  to  draw  the  lines  which 
must  pass  through  the  center?] 

146. 

Cor,    I.     In  how  many  points  can   two    circles 
intersect  ? 
Why?     . 

Exercises. 

72.  A  circle  cannot  have  two  centers. 

73.  If  two  circles  have  a  common  center,  they  must 
coincide. 


CIRCLES. 


93 


74.  How  many  equal  lines  can  be  drawn  from  a  point  to 
a  line  ? 

75.  If  from  any  point  in  a  circle  two  equal  lines  are 
drawn  to  the  circumference,  the  bisector  of  the  angle  formed 
must  pass  through  what  point  ? 

76.  The  right  bisectors  of  the  sides  of  an  inscribed 
polygon  must  pass  through  what  common  point? 

77.  A  radius  perpendicular  to  a  side  of  an  inscribed 
equilateral  triangle  isbisected  by  the  side. 

147. 

Proposition  XI. 

Given  the  equal  circles,  A  and  B  and  the  unequal  chords 
m  n  and  x  y\  ni  n  <  x  y. 

Which  chord  is  nearer  the  center.^ 


(If  unable  to  answer  and  prove,  study  the  figure.  If  still 
unable,  see  the  "hint.") 

[i¥/w/.— Draw  Js  A  C  and  B  D.  Where  are  C  and  D? 
Which  is  longer,  M  C  or  X  D?  Why?  Construct  X  M  == 
M  N.  Draw  1  B  Q! .  Compare  B  C  with  B  D.  Which  is 
greater,  Z  I  or  Z  2  ?  Then  which  is  greater,  Z  3  or  Z  4  ? 
Can  you  now  tell  which  is  greater,  B  D  or  B  C  ?] 

State  and  prove  the  converse  of  the  above.  Write  the 
above  theorem  and  its  converse  as  Prop.  XI. 


94 


PLANE  GEOMETRY.      BOOK  II. 


Exercises. 

78.  Through  a  given  point  within  a  O  construct  the 
shortest  possible  chord. 

79.  Given  two  chords  which  intersect  and  which  make 
equal  Z  s  with  the  radius  drawn  through  the  point  of  intersec- 
tion. Do  the  chords  appear  to  be  equal?  Can  you  prove  them 
equal  or  unequal  ? 

{Hint. — Draw  radii  ±  to  the  chords.] 

80.  In  a  given  ©  construct  a  chord  ||  to  a  given  line  and 
equal  to  another  given  line.  Use  lines  of  different  lengths  and 
in  different  positions.     When  is  this  impossible  ? 

[Hint. — Draw  a  1  from  center  to  the  Ime  to  which  the 
chord  is  to  be  parallel.  Imagine  the  required  chord  drawn, 
and  study  how  to  fix  the  extremity  of  the  chord.] 

If  you  fail  to  construct  the  chord,  study  the  following 
suggestions: 

Given  the  Q.  O,  and  A  B,  the  chord,  and  C  D  the  line  to 
which  the  chord  is  to  be  parallel. 


Imagine  A'  B'  the  required  chord. 

It  is  parallel  to  ^ .     [?] 

It  is  1  to .     [?] 

It  is  bisected  by .     [?] 

Note  that  we  wish  to  discover  how  to  fix  point  b' . 
Imagine  1  B'  N  dropped.     This  1  cuts  off  of  M  D  a  line 
equal  to  ^  of  A  B. 


CIRCLES.  95 

hence  we  can  retrace  the  steps 
taken;  and  we— 

(1)  Bisect  A  B. 

(2)  Measure  M  N  =  >^  A  B. 

(3)  Erect  1  to  C  D  at  N,  which  fixes  B'. 

(4)  Draw  B'  A'  1  to  O  M,  thus  giving  the  required  chord. 
81.     Given    two    equal   intersecting    chords.     Compare 

their  parts. 

148. 
Proposition  XII. 
What  is  a  tangent?  the  point  of  tangency? 


O  M  is  any  radius  of  circle  O.  If  A  B  is  a  line  1  to  O  M 
at  its  extremity, — 

Is  A  B  a  tangent  to  circle  O? 

Prove  your  answer.  State  the  truth  discovered.  State 
and  prove  the  converse.  State  both  as  one  proposition 
(Prop.  XII ). 

(If  you   can  not  prove  Prop.  XII.,  see  the  "hint"  given.) 

\^Hint. — By  the  definition  a  tangent  touches  but  does  not 
enter  the  O-  A  B  does  touch  the  ©,  since  J/  is  the  end  of 
the  radius.  Then  it  is  only  necessary  for  us  to  prove  that 
A  B  will  not  enter  the  ©. 

We  must  show  that  every  point  in  A  B  or  A  B  produced^ 
except  M,  is  without  the  Q. 


96  PLANE  GEOMETRY.     BOOK  II. 

How  does  O  M  meet  A  B  ?  Is  it  then  the  shortest  distance 
from  O  to  A  B  ?  Then  if  any  other  point  in  A  B  than  M,  say 
P,  is  joined  to  O,  the  distance  is  greater  than  the  radius  O  M, 
and  its  end,  which  is  in  A  B,  is  without  the  circle.  But  P  is 
any  other  pt.  in  A  B  than  M;  . ' .  every  point  but  ^is  without 
the  O;  hence  A  B  is  a  tangent  by  definition. 

Note  the  theorem: 

If  a  line  is  perpendicular  to  a  radius  at  its  ex- 
tremity, it  is  a  tangent  to  the  circle. 
Converse: 

A  tangent  to  a  circle  is  perpendicular  to  the 
radius  drawn  to  the  point  of  tangency. 

To  prove  the  converse: 

Draw  a  ©  and  a  tangent.  Draw  a  radius  to  point  of 
tangency. 

Is  the  shortest  distance  from  a  point  to  a  line  a  perpen- 
dicular? Prove  it.  Can  you  prove  this  radius  which  is  drawn 
to  the  point  of  tangency  to  be  the  shortevSt  distance  and  conse- 
quently a  perpendicular? 

When  you  join  any  other  point  in  the  tangent  than  the 
point  of  tangency  to  the  center,  why  is  the  line  longer  than 
the  radius? 

149. 

Cor.  I.  A  perpendicular  from  the  center  of  a 
circle  to  a  tangent  passes  through  the — 

150. 

Proposition  XIII. 

Problem.  To  drazv  a  tangent  to  a  circle  at  a 
given  point  in  the  circumference. 


CIRCLES.  97 

Exercises. 

82.  If  tangents  are  drawn  to  the  extremities  of  a  diame- 
ter, they  are •.     Fill  blank  and  prove. 

83.  If  one  chord  of  one  circle  is  equal  to  one  chord  of 
another,  are  the  circles  necessarily  equal?  If  not,  what  other 
conditions  are  necessary? 

84.  What  is  the  locus  of  the  center  of  a  circle  whose  cir- 
cumference (1)  shall  pass  through  two  given  points? 

(2)  Which  shall  be  tangent  to  two  intersecting  lines? 

151. 

X  circle  is  inscribed  iw  a  polygon  when  each  side  of  the 
polygon  is  tangent  to  the  circle.  The  polygon  is  said  to  be 
circumscribed  about  the  circle. 

152. 

Two  circles  are  coricentric  when  they  have  the  same 
center. 

Exercise. 

85.  Prove  that  the  inscribed  circle  and  circumscribed 
circle  of  an  equilateral  triangle  are  concentric. 

153. 

PtlOPOSITION   XIV. 

I.  Draw  two  parallel  chords  in  a  circle. 

Compare  the  arcs  intercepted  by  the  chords. 

\Hint. — Draw  radius  perpendicular  to  one  of  the  chords 
and  make  deductions.] 

II.  Draw  a  tangent  and  a  chord  parallel  to  it. 

Compare  the  intercepted  arcs. 

III.  Draw  two  parallel  tangents. 

Compare  the  intercepted  arcs. 

State  Prop.  XIV.,  which  includes  the  three  truths 
discovered. 


98  PLANE  GEOMETRY.     BOOK  II. 

154. 

Cor,  Through  what  point  does  the  line  pass 
which  joins  the  points  of  tangency  of  two  parallel 
tangents,  or  the  mid-point  of  a  chord  and  the  point 
of  tangency  of  a  parallel  tangent,  or  the  mid-points 
of  two  parallel  chords  ? 

What  then  is  this  hne  (when  produced  if  necessary)? 
State  corollary. 

[Remember  that  ^z^^rv/>r<?/^/^w,  corollary,  exercise ^  theorem, 
and  locus  requires  geometrical  proofs 

155. 

Proposition  XV. 

Draw  two  circles  which  intersect — (1)  equal  circles;  (2)  un- 
equal circles.  Draw  the  line  of  centers  (line  joining  the  two 
centers).     Draw  the  common  chord. 

(1)  How  does  the  line  of  centers  meet  the 
common  chord  ? 

(2)  Compare  the  segments  of  the  common 
chord. 

Prove  your  answers.     State  Prop.  XV. 

156. 

Proposition  XVI. 
Draw  two  circles  which  are  tangent  externally. 

Does  the  line  of  centers  pass  through  the  point 
of tangency? 

[Hint. — Draw  the  common  tangent  and  erect  1  to  it  at 
point  of  contact.  When  produced  in  both  directions,  through 
what  points  must  the  perpendicular  pass  ?] 


CIRCLES. 

157. 

Proposition  XVII. 


What  is  an  inscribed  angle?  Draw  an  inscribed  angle 
and  the  central  angle  that  intercepts  the  same  arc. 

Compare  the  inscribed  with  the  central  angle. 

(1)  Let  the  drawing  show  one  side  of  the  inscribed 
angle  a  diameter.     Make  deduction  and  prove  this  case. 

(2)  Draw  an  inscribed  Z  wholly  to  the  right  or  to  the 
left  of  the  center.     [Use  the  first  case  in  proving  this.] 

(3)  Let  the  center  be  within  the  inscribed  angle.  State 
Prop.  XVII. 

(If  you  fail  in  the  above  after  studying  the  figure,  making 
drawings  of  your  own,  see  the  "hint"  given  below.) 

\Hint. — See  Figure.  Compare  Z  2  with  Z  3.  Compare 
the  sums  of  Z  2  and  Z  3  with  Z  1.  When  you  attempt  Case 
2,  draw  a  diameter  through  the  vertex  of  the  inscribed  Z  • 
Note  that  the  inscribed  Z  is  diparl  of  an  inscribed  Z  which  has 
a  diameter  for  a  side  and  that  there  are  two  inscribed  Zs 
having  the  diameter  for  a  side.] 


158. 


it. 


Cor.  /.     Draw  a  semicircle  and  inscribe  several  angles  in 
[The  vertex  must  lie  in  the  semi-circumference  and  the 


100  PLANE  GEOMETRY.     BOOK  II. 

sides  of  the  inscribed  angle  must  terminate  in  the  en^ds  of  the 
arc] 

How  large  is  each  angle  inscribed  in  the  semi- 
circle ? 

159. 

Cor.  II.  How  large  is  an  angle  inscribed  in  a 
segment  greater  than  a  semi-circle  when  compared 
with  a  rt.  Z  ? 

160. 

Cor.  III.  How  large  is  an  angle  inscribed  in  a 
segment  less  than  a  semi-circle  when  compared  with 
a  rt.  Z  ? 

161. 

What  can  you  say  of  all  angles  inscribed  in  the  same 
segment  ? 

Exercises. 

86.  What  is  the  sum  of  the  opposite  angles  of  an  in- 
scribed quadrilateral  ? 

87.  Problem. — Erect  a  perpendicular  to  a  given  line  at  a 
given  point  in  it.  [Use  a  different  method  from  that  used  in 
§47.] 

[Hint. — See  §  158.]  [With  center  without  the  line,  draw 
circumference  passing  through  the  required  point.] 

88.  Problem. — Erect  a  perpendicular  to  a  line  from  a 
given  point  withoiit  the  lirie.  [Use  a  different  method  than 
that  in  §  48.] 

[Hi?it. — Draw  any  oblique  line  through  point  to  the  line 
and  use  it  as  a  diameter.] 

89.  Construct  a  right  triangle  when  hypotenuse  and  one 
adjacent  Z  are  given.     [Use  §  158.] 


CIRCLES.  101 

90.  Construct  a  right  triangle  when  hypotenuse  and  one 
leg  are  given. 

91.  Construct  a  right  triangle"  \;s^b,t^n  the  hypotf^ntise  and 
the  altitude  from  the  right  angle  to  the  hypotenuse  are  given. 

92.  From  a  point  without  a  circle  draw  two  tangents. 
[Hi7ii. — Join  point  to  the  center  and  describe   a  circle 

upon  that  line  as  a  diameter.] 
Prove  that — 

(1)  The  tangents  are  equal  in  length. 

(2)  The  line  drawn  from  the  point  without  the  circle  to 
the  center  bisects  the  angle  formed  by  the  tangents  and  also 
bisects  the  central  angle  formed  by  drawing  radii  to  the  points 
of  tangency. 

93.  What  is  the  locus  of  the  vertex  of  a  right  tiiangle 
when  the  hypotenuse  is  given  ? 

94.  Two  chords  are  perpendicular  to  a  third  chord  at  its 
extremities.     Compare  the  two  chords. 


162. 

Proposition  XVIII. 

Draw  any  two  chords  which  intersect  within  the  circle. 

How  large  is  each  vertical  Z  when  compared 
with  the  central  Z  s  which  intercept  the  arcs  which 
subtend  the  vertical  Z  s  ? 

Write  your  answer  as  Prop.  XVIII.  [I^et  the  chords  in- 
tersect in  all  possible  positions  and  draw  the  central  Z  s  and 
compare.] 

[yy/w/.—Show  that  the  required  angle  is  an  exterior  angle 
of  a  triangle  whose  opposite  interior  angles  are  measured  by 
one-half  the  sum  of  the  arcs  which  subtend  the  central  angles.] 


102 


PLANE  GEOMETRY.     BOOK  II. 


163. 
PROi^GSlTlON   XIX. 


Draw  two  secants  intersecting  without  the  circle. 

Compare  the  angle  of  the  secants  with  central 
angles  subtended  by  the  intercepted  arcs. 

\^Hhit. — Compare  Z  1  with  Z  2  and  Z  ^.  But  how  large 
is  Z  2?  how  large  is  Z  3?] 

Write  Prop.  XIX. 

164. 
Pkoposition  XX. 

Draw  a  tangent  to  a  Q  and  any  chord  to  the  point  of 
tangency. 

Compare  the  Z  made  by  the  chord  and  tangent 
with  the  central  Z  which  is  subtended  by  the  inter- 
cepted arc. 


Compare  Z  3  and  Z   1  in  the  figure.     Write  Prop.  XX. 


CIRCLES.  103 

165. 
Proposition  XXI. 

Draw  any  circle  and  a  tangent.  Draw  any  secant  inter- 
secting the  tangent. 

Compare  the  angle  formed  with  the  central  an- 
gles subtended  by  the  intercepted  arcs. 

166. 

Proposition  XXII. 

From  a  point  without  a  circle  two  tangents  are  drawn. 

Compare  the  angle  formed  with  central  angles 
subtended  by  the  intercepted  arcs. 

Exercises. 

94.  Given  two  tangent  circles.  [Draw  in  different  posi- 
tions.]    Draw  the  common  tangent  at  the  point  of  tangency. 

95.  When  are  two  Os  tangent?  externally?  internally? 
Problem. — At  a  given  point  in  the   circumference  of  a 

given  0  draw  a  0  with  a  given  radius  tangent  to  the  given  0. 

96.  What  is  the  locus  of  the  center  of  a  circle  tangentto 
a  given  circle  at  a  given  point  in  the  circumference? 

97.  Given  a  0  with  a  quadrilateral  circumscribed  about 
it.  Compare  the  sum  of  two  opposite  sides  with  the  sum  of 
the  other  two  sides. 

[Hi7it. — Draw  the  figure.  How  many  tangents  or  parts 
of  tangents  have  you?     Which  of  the  parts  are  equal  ?] 

98.  What  is  the  locus  of  the  center  of  a  0  tangent  to 
two  given  lines?    [Suppose  the  lines  were  1 1  ?]     Give  two  cases. 


104 


PLANE  GEOMETRY.     BOOK  II. 


99.  Given  Q  A,  and  A'  B  and  A'  C  two  tangents  from 
any  point  A'  without  the  O;  ^  ^^  is  any  tangent  within  A'  C 
and  A'  B.  What  is  the  sum  of  the  sides  of  the  l\  A!  m  n? 
If  P  moves  on  the  arc  B  C,  will  the  perimeter  of  the  A  in- 
crease or  decrease  ? 

100.  What  is  the  locus  of  the  center  of  a  O  tangent  to  a 
given  line  at  a  given  point? 

101.  What  is  the  locus  of  the  center  of  a  ©of  given 
radius  always  tangent  to  a  given  ©?  Discuss  all  possible 
positions  of  the  tangent  circle  and  the  length  of  its  radius. 

102.  If  two  0s  are  tangent  to  a  line  at  T,  and  if  from  any 
point  (A)  in  the  line  a  tangent  (A  K,  A  ly)  is  drawn  to  each  Q, 
can  you  prove  the  lengths  equal? 


103. 


(1)  Can  you  prove  that  E  D  bisects  m  nf 

(2)  Can  you  prove  m  n  =z  x  yf 

(8)  Can  you  prove  D  E  =  m-7i  =  x yf 

(4)  Can  you  prove  m  C  n  =i  right  Z  ? 

(5)  Can  you  prove  x  C  y  ^=  riRl»t  Z  ? 


CIRCLES.  105 

104.  Construct  a  O  passing  through  a  given  point  and 
also  tangent  to  a  given  line  at  a  given  point.  [What  is  the 
locus  of  the  centers  of  circumferences  passing  through  two 
given  points?  What  is  the  locus  of  the  centers  of  Qs  tangent 
to  a  given  line  at  a  given  point  ?     Will  these  loci   intersect  ?] 

105.  Construct  a  O  tangent  to  a  given  line  at  a  given 
point  and  also  tangent  to  another  given  line.  [May  these  lines 
intersect?  What  is  the  locus  of  the  center  of  a  O  tangent  to 
two  II  lines?  to  two  intersecting  lines ?]     Show  both  solutions. 

(1)  When  the  lines  are  ||. 

(2)  When  they  intersect,  or  when  produced  intersect. 

106.  Construct  a  Q  tangent  to  a  given  ©  at  a  given  point 
and  also  tangent  to  a  given  line.  [Construct  a  tangent  to  the 
given  Q  at  the  given  point.] 


Assume  the  problem  solved  for  the  purpose  of  discovering 
a  solution,  and  assume  that  the  given  circle  is  A,  the  given 
point  is  P,  and  the  given  line  is  m  n,  and  that  the  required  O 
is  B.  With  completed  figure  before  us  it  will  be  easier  to 
discover  a  solution  whereby  we  may  obtain  the  center  B.  Let 
us  try  to  find  two  intersecting  loci  which  will  fix  point  B  for 
us.  Do  you  see  two  given  points  which  joined  and  the  line 
produced  mjist  pass  through  B?  Why  must  it?  State  the 
locus  required.     [Ex.  96  ] 

Is  it  possible  to  draw  another  line  besides  m  nio  which 
O  B  must  be  tangent?     [Ex.  94.] 

Will  this  common  tangent  intersect  m  nt  What  is  the 
locus  of  the  center  of  a  circle  tangent  to  intersecting  lines  ? 


106  PLANE  GEOMETRY.     BOOK  II. 

Must  these  loci  intersect  ?  If  the  common  tangent  and  m  n 
do  not  intersect,  where  must  point  P  lie?  What  is  the  locus 
of  the  center  of  a  circle  tangent  to  two  parallel  lines  ? 

The  above  discussion  is  called  the  analysis  of  the  piob- 
lem.  Having  discovered  the  solution,  we  now  give  the  direct 
construction  as  follows : 

Given:     (1)  Line  m  n. 

(2)  Circle  A. 

(3)  Point  P  in  circumference  of  ©  A. 

Required:  To  draw  a  G  tangent  to  ©  A  at  P  and  to  the 
line  m  n. 

Construction:     (1)  Draw  A  P  and  produce  it. 

(2)  Draw  tangent  to  ©  A  at  P  and  produce  it  until 
m  n  is  intersected  (if  possible). 

(3)  Letter  point  of  intersection  of  tangent  and  m  n,  x^ 
and  then  bisect  /_  m  xY  and  produce  the  bisector  until  A  P 
produced  is  intersected.  Letter  point  of  intersection  B.  Then 
B  is  the  center  of  the  required  ©. 

Proof:  (1)  A  P  produced  is  the  locus  of  "the  center  of 
a  ©  tangent  to  ©  A  at  P.     [Ex,  96.] 

(2)  P  ;i;  is  1  to  A  B.     [§  148.] 

(3)  The  bisector  of  Z  m  xY  \s  the  locus  of  the  center 
of  a  ©  tangent  to  m  n  and  x  P.     [Ex.  84  and  Ex.  98.] 

.  • .   ©  B  is  the  required  ©. 

[Pupil  will  give  solution  when  common  tangent  and  m  n 
are  || ;  also  when  A  P  and  m  7i  are  || ;  also  when  A  P  produced 
(beyond  P)  will  not  cut  m  n.] 

107.  Construct  a  ©  tangent  to  a  given  line  at  a  given 
point  and  also  tangent  to  a  given  ©. 

What  have  we  given?  (1)  A  ©  to  which  the  required© 
is  somewhere  to  be  tangent.  (2)  A  hne  and  the  point  of 
tangency  to  that  line. 


CIRCLES.  •   107 


Again  suppose   the  problem  solved — for   what  purpose? 
Given:     (1)  m  n  the  given  line  and  A  the  given  Q- 

(2)  P  the  point  of  tangency  of  the  required  O  to  that  line. 

(3)  And  B  the  required  Q  (supposed  to  be). 

Again  try  to  find  two  loci  which  intersect  and  thus  fix  B. 
[See  Ex.  106.] 

Can  you  get  the  locus  of  the  center  of  a  ©  tangent  to 
another  ©  ? 

Can  you  get  the  locus  of  the  center  of  a  ©  tangent  to  a 
line  at  a  given  point  ? 

Ca?i  you  get  another  locus  to  intersect  the  locus  found?  Re- 
view the  locii  you  have  learned.     See  Ex.  62,  etc. 

Can  you  apply  Ex,  62  f  (1)  Is  B  equidis  ant  from  the  two 
fixed  points  which  were  given  ?  If  not,  is  it  possible,  with 
data  given,  to  fix  a  point  which  you  can  use  with  P  or  with  A  ? 

[What  straight  line  is  given  ?  What  straight  lines  can 
you  draw  ?  Try  to  fix  a  point  by  measuring  from  fixed  points 
on  given  lines  or  lines  which  you  can  draw.] 

Ex.  62  can  be  used  here.  But  if  it  could  not,  you  should 
try  another  locus,  etc.,  etc.,  till  you  get  a  solution. 


108  ■  PLANE  GEOMETRY.     BOOK  II. 

108.  Construct  a  rectangle  when  the  perimeter  and  the 
diagonal  are  given.  This  is  a  difficult  problem  and  it  is  of 
much  importance  that  the  pupil  fully  weigh  eacl  question  in 
the  order  given,  and  frequently  review  the  ground  gone  over 
to  hold  in  mind  7?^^/  what  has  been  discovered,  that  it  may  be 
readily  used  in  further  investigation. 

Note  carefully  the  data  given  {what  we  know).  Draw  any 
rectangle — suppose  it  to  be  the  required  rectangle  and  study 
it  carefully.  What  data  are  necessary  to  draw  it  ?  Have  we, 
sufficient  data  given  ?  If  not,  how  can  we  by  using  data  given 
get  the  necessary  lines  and  points  to'  draw  it  ? 


Given:     Perimeter 
Diagonal 


:'t}  p^'- 


Or 


How  many  corners  of  the  required  rectangle  do  data  en- 
able us  to  fix? 

If  we  assume  A  as  fixed,  can  we  get  B,  or  the  opposite 
corner,  C?     Can  we  get  the  locus  of  either  of  these  corners? 

What  is  the  locus  of  C  ?  Is  it  a  fixed  distance  from  A  ? 
Do  you  know  that  distance  ? 


CIRCLES. 


109 


Let  us  try  to  discover  another  locus  which  will  enable  us 
to  fix  C. 

[But  why  do  we  want  to  fix  C  ?  How  could  you  draw  the 
rectangle  if  it  were  fixed  ?] 

lyCt  us  now  construct  the  rectangle  so  far  as  possible. 


But  we  must  have  another  locus  of  C,  or  of  D. 

We  have  a  and  also  b;  .' .  we  can  get  a  part  of  them 
should  we  so  desire.  What  use  could  we  make  of  }4  the 
diagonal,  or  of  }4  the  perimeter  ?  Study  the  supposed  rectan- 
gle. You  can  see  there  another  locus  of  C.  What  is  it  ? 
[Circumference  of  a  Q  having  B  the  center  and  B  Cthe  radius.] 
You  no  doubt  think  that  we  have  neither  point  B,  nor  radius, 
B  C,  but  there  is  a  poiftt  in  that  circumference  which  we  ca?i 
fix.  Produce  A  B  till  it  is  the  length  of  the  known  perimeter. 
At  what  point  does  the  above  circumference  cut  A  B  pro- 
duced ?  Can  you  not  fix  that  point  ?  I,etter  it  x.  Note  the 
Z  C  B  ;t:.  What  kind  of  A  would  it  form  the  sides  of?  How 
could  you  draw  at  x  the  third  side  of  the  Ay  or  a  line  that 
must  contain  C,  giving  a  second  locus  of  C  ? 


110  PLANE  GEOMETRY.     BOOK  II. 

109.     Construct  a  Q  with  a  given  radius: 

( 1 )  Tangent  to  two  given  lines. 

(2)  Tangent  to  a  given  line  and  to  a  given  O- 
[Find  intersection  of  loci.] 

(3)  Tangent  to  two  given  Qs. 
[Find  intersection  of  loci.] 

167. 
Proposition  XXIII. 
Problem.      To  circumscribe  a  O  about  a  given  A. 

168. 

Cor.  I.     Pass  the  circumference  of  a  Q  through  three 
given  points.     When  is  this  impossible  ? 

Cor.  II.     Kind  the  center  when  an  arc  is  given. 

169. 
Proposition  XXIV. 

Problem.      To  inscribe  a  Q  in  a  given  A. 

170. 

Cor.  I. — To  draw  a  O  tangent  to  three  given  lines.    When 
is  this  impossible? 

171. 
Proposition  XXV. 
Problem.      To  construct  on  a  given  chord  a  seg- 
ment which  will  contain  a  given  inscribed  Z  . 


Given  chord.  Given  Z  • 

[Use  any  convenient  point  on  either  side  of  Z  ^  as  a  cen- 
ter and  with  a  for  radius  cut  the  other  side  of  Z  b.     Pass  a  cir- 


CIRCLES.  HI 

cumference  through  the  three  points  fixed.  Pupil  should  note 
the  segment  and  give  the  geometrical  proof.  Prove  that  any 
Z  inscribed  in  this  segment  equals  Z  ^.] 

Exercises. 

/ 

109.  To  construct  a  A  when  the  base,  altitude,  and  Z 

of  vertex  is  given.     [Use  the  above  problem.] 

110.  To  construct  a  A  when  two  sides  and  the  Z  op- 
posite one  side  are  given.  Show  solution  when  the  Z  is 
(1)  acute,  (2)  right,  (3)  obtuse. 

111.  (1)  To  construct  a  A  when  the  base,  the  vertical  Z  , 
and  the  median  from  base  are  given.  When  is  this 
impossible  ? 

(2)  What  is  the  locus  of  the  vertex  of  a  A  when  the 
base  and  the  vertical  Z  are  given? 

112.  Problem.  To  draw  a  common  tangent  to  two  given 
circles. 


How  many  common  tangents  may  be  drawn  to  these  Qs? 
How  many  common  tangents  cross  the  line  of  centers? 

[Join  O  to  O'.  With  O  for  center,  draw  a  Q  with  radius 
R — r.  From  O'  draw  a  tangent  to  this  Q-  r)raw  its  radius  to 
the  point  of  tangency  and  continue  to  the  circumference  of 
large  O-  At  this  point  draw  a  tangent  to  large  O  O.  Can 
you  prove  this  line  also  a  tangent  to  Q  O'J*  ^^y  to  invent  a 
way  to  draw  the  tangents  which  cross  the  line  of  centers.] 


112  PLANE  GEOMETRY.     BOOK  II. 

KXERCISRS. 

113.  Can  you  find  the  center  of  a  ©  without  bisecting 
any  straight  line? 

114.  Inscribe  a  rectangle  in  a  ©  using  diameters  only 
to  fix  points. 

115.  Draw  a  O  and  take  any  point  in  it.  What  is  the 
longest  possible  line  that  can  be  drawn  from  this  point  to  a 
point  in  the  circumference? 

116.  Draw  a  Q  and  fix  any  point  without  it.  Draw  the 
shortest  possible  line  from  the  point  fixed  to  the  circumfer- 
ence of  the  circle. 

117.  What  is  the  locus  of  the  mid-point  of  a  chord  of  a 
given  length  in  a  given  circle. 

118.  Can  a  tangent  of  any  kind  be  drawn  to  a  Q  from  a 
point  within  it? 

119.  Can  a  tangent  be  drawn  to  a  point  in  the  circum- 
ference when  the  center  is  not  known? 

120.  Describe  a  circle  on  one  of  the  sides  of  an  isosceles 
A  and  show  how  the  circumference  cuts  the  base. 

121.  Find  the  locus  of  the  mid-point  of  a  ladder  as  the 
foot  of  the  ladder  is  pulled  away  from  a  vertical  wall. 

122.  Through  a  given  point  within  an  angle  draw  a  line 
intercepted  between  the  sides  and  bisected  at  the  given  point. 
[See  Ex.  107  and  Ex.  108  for  method;  and,  if  you  fail,  see 
§  118.] 

123.  Construct  an  equilateral  A  when  the  altitude  is 
given. 

124.  Trisect  a  given  rt.  Z  . 

125.  A  circle  is  wholly  without  or  within  another  circle, 
according  as  their  central  distance  is  greater  than  the  sum,  or 
less  than  the  difference,  of  their  radii. 

[From  Ex.  125  cm  you  show  how  the  central  distance  is 
related  to  the  radii  if  two  Qs  intersect?] 


MEASUREMENT.  113 


BOOK    III. 

MKAvSUREMENT. 
172. 

Plane  Geometry  deals  with  lines,  angles,  surfaces,  and 
areas  of  surfaces.  In  comparing  these  magnitudes  hitherto 
it  has  been  deemed  sufficient  to  prove  their  equality,  or  to 
prove  that  one  is  greater. 

[When  it  is  known  that  two  Z  s  of  a  A  are  unequal,  what 
follows  ?  If  in  the  same  ©  or  in  equal  Qs  chords  are  unequally 
distant  from  the  center,  what  relation  do  ihese  chords  have  ? 
Can  you  think  of  other  propositions  wherein  lines  have  been 
proved  unequal  ?  Think  of  propositions  in  which  Z  s  have 
been  proved  unequal,  etc.,  etc  ] 

But  it  now  becomes  necessary  for  us  to  find  the  exact  re- 
lation of  the  size  of  two  or  more  magnitudes. 

Case  I. — If  in  comparing  the  lengths  of  two  lines,  a  and  b, 
we  find  that  b  contains,  or  measures,  a  an  exact  number  of 
times,  say  three  times,  what  is  the  relation  of  ^  to  a  ?  of  «  to  ^  ? 

Case  II. — If  in  comparing  the  lines  a  and  b  we  find  that 
one  is  not  an  exact  measure  of  the  other,  but  that  a  third  line, 
c,  is  exactly  contained  in  a  four  times  and  in  b  exactly  nine 
times,  what  is  the  relation  of  «  to  ^?  of  ^  to  a?  What  is  their 
common  measure?  Can  you  tell  the  common  measure  in  Case 
I.?  When  is  one  line  a  common  measure  of  another?  Sup- 
pose a  and  b  are  given  and  ^is  required.     Can  you  find  ct 

[^ow  did  you  find  the  highest  common  factor  of  two  or 
more  quantities  in  Arithmetic  and  Algebra  when  you  were 
unable  to  factor  the  quantities  ?] 


114 


PLANE  GEOMETRY,  BOOK  III. 


Given  the  lines  «, 


\,  and  dy 


It  is  required  to  find   their   common   measure.     Apply  the 
smaller,  a,  to  the  larger,  d. 


We  find  a  is  contained  three  times  in  d  with  a  remainder, 
r.     Apply  r  to  a.  /** 

[Why  is  r  less  than  a  ?]  ''      '' 

We  find  r  is  contained  in  a  once  with  a  remainder,  r. 
Apply  /  to  r.  r 


We  find  /  is  contained  in  r  twice  and  l/iere  is  ?io  remain- 
der. What  must  be  added  to  r  to  produce  «?  «  is  then  how 
many  times  r'^.b  is  also  how  many  times  /-'?  What  then  is 
the  common  measure  oia  and  ^?  What  is  the  relation  oi  b  to 
a  ?  of  a  to  ^  ?     What  is  the  relation  of  r  to  «  ?  to  3  ?  to  /  ? 

Case  III. — Let  us  try  to  find  a  common  measure  of  the 
diagonal  and  a  side  of  a  square. 


Let  A  B  C  D  be  the  required  square.  Draw  the  diagonal 
C  A.  Apply  C  B  to  C  A  (C  B  <  C  A;  [Auth.]  C  A  is  not 
twice  C  B;    [Auth.]     .*.  C  B  is  contained  in  C  A  once  with  a 


MEASUREMENT.  115 

remainder.  Lay  oflf  on  C  A,  C  P  equal  to  C  B;  then  the  remain- 
der is  A  P.  Erect  a  1  to  C  A  at  P,  which  cuts  A  B  at  N. 
then  B  N  =  N  P;  [Auth.]  also  N  P  =  A  P,  and  A  P  N  is  a  rt. 
isos.  A,  and  A  P  N  M  is  a  square.  [Give  proof  of  each  state- 
ment.] When,  then,  we  apply  the  remainder,  A  P,  to  a  side, 
C  B,  or  to  its  equal,  A  B,  A  P  is  contained  twice  with  the  re- 
mainder N  P'.  Show  that  N  P'  (second  remainder)  is  con- 
tained in  A  P  (first  remainder)  twice  with  a  remainder,  and 
eonsequently,  show  there  will  always  be  a  remainder  and  that 
there  is  no  common  measure  of  the  diagonal  and  a  side  of  a 
square. 

Such  lines  are  said  to  be  incommensurable. 
Do  you  know  of  other  lines  that  are  incommensurable? 
While  it  is  impossible  to  get  the  exact  relation  of  two 
incommensurable  magnitudes,  we  may  approximate  that  re- 
lation to  any  required  degree  of  accuracy.  '  (1)  Thus,  when 
C  B  is  divided  into  10  equal  parts,  [Review  glH.]  A  C  con- 
tains more  than  14  of  those  parts  and  less  than  15.  (2)  If 
C  B  is  divided  into  100  parts,  A  C  will  contain  more  than  141 
and  less  than  142  of  them,  etc. 

In  the  first  instance  A  C  is  more  than  \%  and  less  than  \% 
times  B  C. 

Tell  the  relation  of  A  C  to  B  C  in  the  second  instance. 
Tell  the  relation  of  B  C  to  A  C  in  each  instance. 

Extract  the  square  root  of  2  true  to  six  decimal  places 
and  further '•approximate  the  relation  of  A  C  to  B  C.  Can 
you  in  each  instance  find  two  lines,  one  a  little  less  than  A  C 
and  one  a  little  more  than  A  C,  to  which  B  C  does  bear  an 
exact  relation?  Find  the  difference  of  these  two  lines  in  each 
instance.  Do  these  lines  get  closer  and  closer  to  A  C,  and 
does  their  difference  grow  less  and  less?  What  is  the  differ- 
ence when  you  use  all  of  the  six  decimal  places  required  above? 


116  PliANE  GEOMETRY,  BOOK  III. 

RATIO. 
173. 

What  is  the  relation  of  line  a  to  line  b  when  a  is  2  ft.  in 
length  and  <^  is  9  ft.?  when  «  is  3  yds.  and  b  is  2  rods?  When 
a  is  2J  inches  and  b  is  2^  ft.? 

The  fraction  or  quotient  which  expresses  the  relation  of 
two  magnitudes  is  called  the  ratio  of  those  magnitudes.  Is  it 
possible  to  find  the  ratio  of  two  magnitudes  of  different  kinds? 
What  is  the  ratio  of  9  ft.  to  10  yds.?  of  40  rods  to  2  miles?  of 

1  of  an  inch  to  2J  ft.?     Can  you  find  the  ratio  of  2  inches  to 

2  sq.  ft.?  of  3 J  sq.  yds.  to  50  sq.  ft.? 

Define  ratio. 

The  ratio  of  a  quantity  «  to  a  like  quantity  b  is  expressed 

a  \  b,  or  -,  and  is  read  in  each  case,  the  ratio  of  a  to  b.     a  is 
b 

the  first  term  of  the  ratio,  or  the  antecedent,  and  b  is  the  second 

term,  or  co?iseque7it. 

When  the  diagonal  of  a  square  is  1  foot,  write  the  frac- 
tion that  expresses  the  ratio  of  the  diagonal  to  the  side.  Is 
the  fraction  a  rational  fraction?  [See  Algebra  for  definition 
of  rational^  Write  the  rational  fraction  that  is  less  than  one- 
millionth  greater,  and  also  the  rational  fraction  that  is  less 
than  one-millionth  less  than  the  required  ratio  when  the  side 
of  the  square  is  1  foot. 

When  two  incommensurable  quantities  of  the  same  kind 
are  given,  is  it  always  possible  to  find  the  ratio  within  any 
required  degree  of  accuracy? 

When  the  side  of  a  square  is  1  yard,  find  two  fractions 
each  of  which  approximates  the  ratio  of  the  diagonal  to  the 
side  within  one  ten-thousandth;  within  one  ten-millionth. 

I^et  A  and  B  be  two  incommensurable  quantities.  Let 
A  be  divided  into  any  number  of  equal    parts,  say  n  equal 


RATIO.  117 

parts,  and  let  P  be  each  part.  Then  A  =  ?  Then  B  must 
contain  some  number  of  those  parts,  say  m  parts,  with  a  re- 
mainder less  than  P.  Then  B  is  greater  than  m  P,  [Why?] 
and  less  than  (/«+l)P.     [Why?] 

If  it  is  required  to  approximate  the  ratio  of  B  to  A,  we 
have 

B        Pw        .         Vim^X)         B        m       ^        m-\-\ 

-r  >  TT-  ana   <  -^^ir ;  or  —  >  —  and  < . 

A        Vn  ^       F?i  An  ^      n 

How  much  srreater  than  —  is ? 

°  n  n 

But  A  was  divided  into  n  equal  parts;  so  if  we  make  n  a 

tn 
very  great  number,  the  diflference  between  the  ratios  —  and 

n 

— ■' — ,  which  is  always  -,  will  become  very  small.     What,  then, 

71  71 

can  you  say  of  the  ratio  of  two  incommensurable  numbers? 

{Hint.  —The  pupil  must  thoughtfully  read  each  question 
in  the  order  asked.     Master  it.] 

Exercises. 

126.  What  is  the  ratio  of  1  lb.  to  9  ozs.?  to  32  ozs.?  to 
2J  lbs.?  to  3f  lbs.?  to  ^  ozs.? 

127.  What  is  the  ratio  of  90°  to  2^3^  st.  Z  s? 

128.  If  a  base   Z  of  an  isosceles  A  is  40°,  what  is  its 
ratio  to  the  vertical  Z  ? 

129.  If  the  vertical  Z  of  an  isos.  A  is  42°,  what  is  its 
ratio  to  each  base  Z  ? 

130.  If  the  exterior  angle  at  the  base  of  an  isos.  A  is 
110°  degrees,  what  is  its  ratio  to  each  of  the  interiorr  Z  s-* 


118  PLANE  GEOMETRY,  BOOK  III. 

PROPORTION— PRBI.IMINARY. 
174. 

What  is  the  ratio  of  2  feet  to  3  inches  ?  of  13 1^  yards  to 
5  feet?  What  can  you  say  of  these  two  ratios?  What  num- 
ber has  the  same  ratio  to  10  feet  that  160  rods  has  to  one 
mile  ?  What  two  numbers  have  the  same  ratio  to  each  other 
that  3  a  has  to  7  a?  that  5  feet  has  to  4  rods?  When  two 
pairs  of  quantities  have  equal  ratios,  they  are  said  to  be  in 
proportion.  Thus  if  it  is  known  that  the  ratio  of  A  to  B 
equals  the  ratio  oi  x  to  j/,  the  four  quantities  are  inproportioji, 

A        X 

and  may  be  written  —  =:-,orA:B=;i::jK,  and  each  is  read 
£>        y 

the  ratio  of  K  to  ^  equals  the  ratio  of  x  to  y\  or  each  may  be 
read  A  is  to  B  as  x  is  to  y.  Which  term  is  the  antecedent  in 
any  ratio?  the  consequent?  What  are  the  antecedents  ia  the 
above  proportion?  the  consequents? 

175. 

(1)  The  extremes  in  a  proportion  are  the  first  and  fourth 
terms;  (2)  the  means  are  the  second  and  third  terms;  (3)  when 
three  numbers  are  in  proportion,  2.?,  a  \  b  ^^^  b  \  c\  b  is  called  a 
mean  proportional  and  c  is  called  a  third  proportional. 

Can  there  be  a  ratio  between  two  unlike  magnitudes? 
Can  there  be  a  proportion  when  the  four  quantities  are  not 
all  the  same  kind  ? 

[Give  examples  of  youi  answers.] 

176. 

In  ratio  we  may  measure  one  magnitude  by  another  mag- 
nitude; e.  g.,  the  ratio  of  line  a  to  line  b  equals  |  when  the 
lines  a  and  b  contain  a  common  measure,  c,  4  and  9  times,  re- 

«       4  4 

spectively,  and  we  write  7  =  rr-     We  call  7^  the  numerical  ratio 

o      V  y 

of  a  to  b. 


I'ROPORTION.  119 


177. 

Suppose  we  wish  to  give  the  geometrical  proof  of  the 
above,  which  stated  is — 

The  ratio  of  two  magnitndes  is  cqiial  to  the  ratio  of  their 
numerical  measures  when  referred  to  the  same  unit. 

We  have  «  =  4  <:;     [Hyp.] 

Also  (^  =r  9  ^;     [  ?  ] 

•    •    ^  -  9  ^'     L  •  J 

^  c         4       r  ■%  T 

•     -=-•    [M 

[Hint. — To  make  this  proof  general  substitute  m  for  4 
and  n  for  9.] 

178. 

Theorem.  Prove  that  if  four  magnitudes  are  in  propor- 
tion, their  numerical  measures  are  in  proportion  and  con- 
versely. 

[Hint. — I^et  the  four  magnitudes  be  A,  B,  C,  and  D,  and 
their  numerical  measures  be  a,  by  c,  and  d,  respectively. 

Then|  =  f.    in 


Also|=^,    [?]     etc.] 

[Pupil  will  finish  the  above  proof.] 


120  PLANE  GEOMETRY,  BOOK  III. 

PROPORTION.  -  Propositions. 
Preliminary  Questions. 

1.  Observe  that  when  four  numbers  are  in  proportion, 
we  have  two  equal  fractions.  Select  two  equal  fractions  and 
make  all  the  deductions  you  can. 

(1)  Are  the  results  equal  when  the  fractions  are  in- 
verted? When  the  denominators  are  interchanged  ?  When 
the  denominator  of  the  first  is  interchanged  with  the  numer- 
ator of  the  second?  etc.,  etc. 

(2)  Are  their  powers  equal ?  their  roots? 

(3)  Does  the  sum  of  the  numerators  divided  by  the  sum 
of  the  denominators  equal  each  of  the  fractions  ?  Is  this  true 
when  there  are  three  or  more  equal  fractions  ? 

(4)  Is  the  sum  of  the  terms  of  the  first  divided  by  either 
term  equal  to  the  sum  of  the  terms  of  the  second  divided  by 
the  corresponding  term?  etc.,  etc. 

(5)  May  the  numerators  or  the  denominators  be  multi- 
plied or  divided  by  the  same  number  and  the  results  remain 
equal  ? 

2.  (1)  May  four  magnitudes  be  in  proportion  when  they 
are  like  magnitudes?  e.  g.,  four  rectangles?  four  solids? 

(2)     May  four  magnitudes  be  in  proportion  when  the 
magnitudes  are  not  all  of  the  same  kind? 
[Hint. — Carefully  review  the  definition  of  a  proportion.] 

179. 

Proposition  I. 

Select  four  numbers  which  are  in  proportion. 

Does  the  product  of  the  extremes  equal  the  prod- 
uct of  the  means? 

lyCt  the  numbers  be  unknown  to  you;  e.  g.,  a,  b,  c,  and  d, 
and  \^i  a  \  b  ^=  c  :  d.  What  is  the  question  you  wish  to 
answer  about  these  four  numbers?     What  other  way  to  write 


PROPORTION.  121 

the  proportion?  By  a  single  operation  you  can  obtain  the 
desired  proof.     Try  to  discover  it.     What  axiom  do  you  use? 

Select  four  magnitudes  which  are  in  proportion.  Is  it 
possible  to  multiply  the  extremes  together?  Does  the  above 
truth  apply  to  magnitudes  when  they  are  all  of  the  same 
kind?  when  the  pairs  of  magnitudes  are  of  different  kinds? 
Does  it  ever  apply  to  magnitudes? 

[N.  B. — The  above  is  no  part  of  Xh^  formal  proof ,  which 
it  is  desired  the  pupil  shall  originate  for  himself.  These 
questions  are  asked  to  lead  him  to  discover  the  truth  and  its 
proof  .^ 

The  proof  of  Prop.  I.  is  given  below  to  serve  as  an  ex- 
ample of  proofs  in  Proportion. 

I.  If  four  numbers  are  in  proportion,  the  product  of  the 
extremes  equals  the  product  of  the  means. 

Given:     a  \  b  -=^  c  \  d. 
To  prove:     ad=^  be. 

Proof:     ?  =  £;     [Hyp.] 

.*.  ad  =^  be.     [Ax.  If  the  same  operation  be  performed 
upon  equals,  the  results  will  be  equal.] 
[What  was  done  to  each  fraction?] 

II.  If  four  li7ies  are  in  proportion,  the  rectafigle  of  the 
extremes  equals  the  rectangle  of  the  means. 

[flint. — Let  the  magnitudes  be  represented  by  capitals 
and  their  numerical  measures  represented  by  small  letters.] 

Given:     A  :  B   =  C  :  D. 

Then  a  :  b  =  c  \  d.     [Prop.  When  four  magnitudes  are  in 
the  proportion,  their  numerical  measures  are  in  proportion.] 
a  c 

.     "^'b^d'^ 

.'.     ad  =  be.     [?] 

But  a  d^  represents  the  product  of  the  numerical  measures 
of  the  lines  A  and  D,  or  the  rectangle  formed  by  A  and  D; 


122  PI^ANE  GEOMETRY,  BOOK  III. 

also  be  represents  the  product  of  the  numerical  measures  of 
B  and  C. 

.•.     AD  =  BC.     [Which  means  the  rectangle  formed  by 
A  and  D  equals  the  rectangle  formed  by  B  and  C] 

180. 

Proposition  II. 

What  is  the  converse  of  Prop.  I.?     Write  the  converse  to 
each  part  and  give  the  formal  proof. 

181. 

Proposition  III. 

I.     Write  two  or  more  proportions. 

Are  the  products  of   the   corresponding  terms  in 
proportion? 

Does  your  answer  apply  to  lines  as  well  as  to  numbers? 
Write  Prop.  III.  and  its  proof. 

182. 

Proposition  IV. 

When  four  numbers  are  in  proportion,  are  like 
powers  or  like  roots  of  these  numbers  in  proportion? 

Write  Prop,  IV.  and  its  proof. 

Can  this  proposition  be  applied  to  lines  or  other  geomet- 
ric magnitudes? 


TROPORTION.  123 

183. 

Proposition  V. 

If  four  like  quantities  are  in  proportion,  «^^  they 
in  proportion  by  alternation?  that  is,  is  the^rst  to  the 
third  as  the  second  is  to  the  fourth? 

Is  this  true  when  the  pairs  of  quantities  are  unlike  ? 
Write  and  prove  Prop.  V- 

[flint. — Let  us  attempt  the  proof  when  the  quantities 
are  like. 

Given:     I.     A,  B,  C,  and  D.  the  like  quantities. 

2.  a,  b,  c,  and  d,  their  numerical  measures. 

3.  Also,^  =  |. 

To  prove:     ^  ==  g- 

When  we  can  change  what  is  given  by  the  hypothesis 
directly  into  the  required  statement,  we  have  what  is  called 
the  direct  proof .  But  it  is  often  difficult  to  see  the  steps  nec- 
essary to  do  this,  and  we  will  now  outline  a  method  whereby 
we  are  often  enabled  to  discover  the  necessary  steps. 

1st.  V\yL  firmly  in  mind  what  is  given;  this  we  know  to 
be  true. 

2d.     Clearly  fix  in  mind  what  is  required  to  be  proved. 

3d.  Suppose  that  which  we  wish  to  prove  is  really  proved, 
and  then  make  deductions  based  upon  the  truth  of  this  sup- 
position. After  each  deduction,  try  to  find  a^r^<?/"for  the  state- 
ment made;  if  you  are  unable  to  discover  a  proof,  make 
further  deductions  and  again  try  to  prove  each  in  turn. 
When  finally  able  to  prove  the  last  deduction  made,  prove  the 
one  immediately  preceding  the  last  by  using  the  last  deduction 
and  continue  to  retrace  the  steps  taken,  proving  each  in  re- 
verse order  until  the  supposed  truth   is  finally  established, 


124  PLANE  CEOMETRY,  BOOK  III. 

based  upon  a  chain  of  evidence  from  the  last  to  the  first;  e.  g., 
in  Prop.  V.  we  know 

A        C 

:p  =  rr,  and  we  wish  to  prove  that 

A  ^  B 
C  ~  D' 
Now  apply  the  method  outlined. 

A        B 

(1)      p  =  K-     [Supposed  to  be  true.] 

(9.\     Th       ^  ^       [First  deduction.] 

c        d'     [When     four     magnitudes     are     in 
proportion ] 

(3)  And  a  d^=^  b  c.     [  ?  ]     [Second  deduction.] 

(4)  ^  =  ^  -1 

^  '     bd        bd'      '-■  -■  ,^, , 

\       [Third  deduction.] 

Or      ^  -    ^-  ! 

A         C 

(5)  .'.  —  =  —.     [  ?  ]     [Fourth  deduction.] 

But  we  know  that  this  fourth  deduction  is  true  by  hypothe- 
sis: consequently  we  have  discovered  the  steps  whereby  we 
may  give  the  direct  proof,  which  is: 

A        C 

1.  —  =  —  .     [Hyp.]     {^Fourth  deduction.] 

2.  Then  ^  ==  ^-     [  ?  ]     [  Third  deduction.] 

3.  a  d  —  b  c.     [  ?  ]     {^Second  deduction  ] 

4.  -  1=:  -  .     [  ?  ]     {^First  deduction.] 

A        B 

5.  .•.—==—..     [  ?  ]     [The  required  equality.] 


PROPORTION.  125 

184. 

Proposition  VI. 

Are  four  numbers  which  are  in  proportion  in 
proportion  by  inversion;  that  is,  is  the  second  to  the 
first  as  the  fourth  is  to  the  third? 

Are  four  magnitudes  which  are  in  proportion  in  propor- 
tion by  inversion  f 

(1)  When  the  magnitudes  are  like  f 

(2)  When,  the  pairs  of  magnitudes  are  unlike  ? 

Do  you  conclude  that  ajiy  four  quantities  which  are  in 
proportion  are  in  proportion  b}^  inversion  ? 

Try  to  prove  it  is  true. 

lyCt  A,  B,  C,  and  D  be  the  four  magnitudes,  and  a,  b,  c, 
and  d  be  their  numerical  measures. 

Then  we  have: 


Given: 

A  _ 
B  ~ 

C 
D' 

To  prove:     -^ 

D 

~  c  • 

Proof: 

A 
B  ~" 

-     [M 

Then^, 

0 

c 
~  ~d' 

[?] 

lyct  the  pupil  finish  the  proof  an 

d  write 

carefully  Prop. 

VI. 

185. 

Proposition 

vir 

By  composition  is  meant  the  sum  of  the  fivst  and 
second  is  to  either  the  first  or  to  the  second  as  the  sum 
of  the  third  and  fourth  is  to  either  the  third  or  to  the 
fourth. 

Ask  a  comprehensive  set  of  questions  with  respect  to 
composition  about  four  quantities  which  are  in  proportion  [see 


126  PLANE  GEOMETRY,  BOOK  III. 

questions  under  Prop.  VI.]  and  give  their  answers  and  care- 
fully write  the  formal  statement  of  Prop.  VII.  and  give  its 
proof. 

186. 

Proposition  VIII. 

By  division  is  meant  the  difference  of  the  first 
and  the  second  is  to  either  the  first  or  the  second  as 
the  difference  of  the  third  and  fourth  is  to  the  third 
or  to  the  fourth. 

Treat  division  in  a  similar  manner  to  that  of  composition 
in  Prop.  VII. 

187. 

Proposition  IX. 

When  four  numbers  are  in  proportion,  are  they 
in  proportion  by  composition  and  division;  that  is, 
the  sum  of  the  first  and  second  is  to  their  difference 
as  the  sum  of  the  third  and  fourth  is  to  their  dif- 
ference ? 

Is  this  true  when  four  like  magnitudes  are  in  proportion  ? 
when  the  pairs  are  wiLike  ? 

Carefully  study  all  possible  kinds  of  proportions  with 
reference  to  composition  and  dlvlslo?i  and  then  state  and  prove 
Prop.  IX. 

Exercises. 

131.  li  a  \  b  =^  c  :  d,  prove  that  a  \  a  -\-  b  ^=  c  :  c  -\  d 
Is  this  true  when  a  and  b  are  unlike  c  and  d} 

132.  li  a  \  b  ^=^  c  '.  d,  prove  that  a  —  b  :  c  —  d  =z  b  :  d. 
May  a,  b,  c,  and  d  represent  any  quantities  whatever?  State 
clearly  the  exceptions  if  there  are  any. 


PROPORTION.  127 

188. 
Proposition  X. 

1.  Write  a  series  of  equal  ratios,  (I)  using  all  numbers, 
(2)  using  lines,  (3)  using  other  like  quantities,  (4)  using  unlike 
quantities,  if  possible. 

In  each  case  the  quantities  used  are  said  to  be  in  continued 
proportion. 

Carefully  examine  each  continued  proportion  and  answer 
the  following  question  about  it :  Is  the  sum  of  all  the  ante- 
cedents to  the  sum  of  all  the  consequents  as  any  antecedent 
in  that  continued  proportion  is  to  its  consequent  ? 

If  your  continued  proportions  comprehend  all  possible 
cases,  you  should  now  be  able  to  state  Prop.  X.  Try  to  prove 
your  statement. 

[Hint. — Let  A,  B,  C,  D,  E,  F,  etc.,  be  any  like  quantities 
and  A:B  —  C:D=:E:F,  etc  Then  a  \  b  —  c  \  d  =  e  \  f, 
etc.     [Auth.] 

^     a         c         e 

^'^J=d  =  f  =  ^'''' 

To  prove:  A  +  C+E+....:B  +  D  +  F+  ••••  = 
A:B  =  C:D=E:F 

(1)  li\=r. 
Then  ~  =z  r,  [  ?  ] 

And  T'  =  ^,     [  ?  ]     etc. 

(2)  :•.  a=br     [?] 
c=dr,     [?] 

e  ^^  fry     [  ?  ]     etc. 

(3)  Anda  +  ^  +  ^+  ....  =  (^4-^+./+  ....)^     [?] 

Pupil  will  finish  the  proof.] 


128  PLANE  GEOMETRY,  BOOK  HI. 

189. 

Proposition  XI. 

1.  Write  a  proportion  (1)  using  numbers,  (2)  using  lines 
or  other  like  magnitudes,  (8)  using  magnitudes  having  the 
pairs  unlike. 

2.  (1)  Will  the  numbers  or  magnitudes  be  in 
proportion  when  the  antecedents  are  multiplied  by 
the  same  number?  (2)  When  the  consequents  are 
multiplied  by  the  same  number  ?  (3)  When  the 
antecedents  are  multiplied  by  one  number  and  the  con- 
sequents by  another  nuTnber  f 

State  and  prove  Prop.  XL 

190. 
Proposition  XI I. 

Is  the  value  of  a  ratio  changed  when  both  terms 
are  multiplied  or  divided  by  the  same  number  ? 

State  and  prove  Prop.  Xll. 

191. 

Cor.  If  A  :  B  =  P  :  Q  when  A  and  B  are  like  quantities 
and  P  and  Q  are  also  like  quantities, 

(I)     IsM  A  :  M  B=  N  P  :  NQ? 

State  and  prove  the  corollary. 

Exercises. 

133.  \{a\b  =  b\x,^x\^x. 

134.  \i  a  \  x^=^  X  :  c,  find  x. 

135.  \i  a  '.  b  =^  c  \  X,  find  x. 


PROPORTION— LIMITS.  129 

136.     Ua  :  d  =  c  :  d,  and 
a  :  d  =^  e  :/. 
Show  that  <:  :^  =:<?: /. 
J37.     If  A,    B,  C,   D,  H.   F,  etc  ,  are  like  quantities,  and 

A:  B  =  C:D=  E:F== ,  and  rn,  ?i,  o,  p,  etc.,  are  numbers, 

Am  +  C^^  +  E^+   ..■■  _  A  _  C  _  E  _ 
^"b  ;«  +  D  ;2  -h  F  ^+  ....        B       D       F 

192. 

Proposition  XIII. 

Given  a?iy  two  proportions  in  which  three  terms  of  one 
are  respectively  equal  to  three  terms  of  the  other ;  can  you 
prove  that  the  fourth  terms  are  equal  f 

[The  proof  should  comprehend  all  possible  proportions.] 


LIMITS. — Preliminary  Questions. 

193. 

[The  pupil  should  carefully  review  measuremeiit  under 
Proportion.] 

1.  \im  and  7i  are  two  points  a  given  distance  apart  and 
A  move  from  m  directly  toward  7i  and  travel  one-half  of  the 
distance  from  m  to  7i  the  first  day,  and  then  one-half  the  re 
maining  distance  the  second  day,  and  then  one-half  the  dis- 
tance still  rermining  the  third  day,  etc.,  will  A  ever  reach  ;^? 

2.  Draw  a  diagram  showing  A  at  the  end  of  each  of  the 
three  days;  also  show  the  position  of  A  at  the  end  of  the  fourth 
day. 

3.  What  can  you  say  of  the  distance  from  m  to  A  at  the 
end  of  each  succeeding  day  ?  What  is  the  limit  of  that  dis- 
tance, or  the  distance  which  m  A  can  7iever  reach? 

4.  What  can  you  say  of  the  distance  from  n  to  A  from 
day  to  day?     Has  ^^  A  a  limit  which  it  can  never  reach  ? 

5.  Would  you  call  m  A  and  7i  A  variable  quantities  or 


130  PLANE  GEOMETRY.  BOOK  III. 

variables?  Can  you  distinguish  between  them  and  again  tell 
the  limit  of  each? 

6.  Does  the  distance  which  A  travels  also  vary  from  day 
to  day?  Name  three  variables  in  the  preceding  discussion. 
Which  two  of  the  variables  are  of  the  same  kind?  What  is 
the  limit  of  each  f 

1.  If  A  travel  a  uniform  distance,  say  c,  each  day,  would 
you  call  e  a  variable  f  Would  A  ever  reach  n  ?  How  many 
days  are  required  for  A  to  reach  n  ? 

8.  Define  a  variable;  a  constant.  Do  you  think  m  n\s  2i 
constant  f 

\^Hint. — Does  its  value  remain  fixed  during  this  entire 
discussion?] 

What  else  have  you  called  mnm.  this  discussion  ? 

9.  As  another  illustration,  consider  the  series  1,  4,  \,  ^, 


-lV»  ¥V' 


What  kind  of  variable  is  the  last  term  of  the  series? 

[  (1)  Think  of  the  series  having  two  terms,  (2)  three 
terms,  {3)  four  terms,  etc.] 

What  does  it  approach  as  a  limit  f 

What  is  the  sum  of  the  terms  of  the  series  when  three 
terms  are  taken  ? 

What  is  the  sum  of  the  terms  of  the  series  when  four 
terms  are  taken  ? 

What  is  the  sum  of  the  terms  of  the  series  when  five 
terms  are  taken  ? 

What  is  the  sum  of  the  terms  of  the  series  when  six 
terms  are  taken  ? 

What  is  the  sum  of  the  terms  of  the  series  when  seven 
terms  are  taken  ? 

What  other  variable  do  you  discover  ?     What  is  its  limit  f 

10.  Given  the  line  m  7i\  bisect  it  at  A;  bisect  A  ?2  at  A'; 
bisect  A'  n  at  A",  etc.  Let  x  represent  the  distance  from  m  to 
any  of  the  points  A  A',  A",  etc.     What  do  you  call  r?     Why? 


PROPORTION— LIMITS.  131 

Does  the  value  oimn  change  in  this  discussion  ?  Call  m  n,  a. 
What  is  «  ?     What  does  x  approach  as  a  limit  f 

(The  sign  =  means  approach  as  a  limit.) 

We  write  x  ==.  a.     Read  the  statement. 

Is  a  —  X  2i  variable  ?  What  does  it  approach  as  a  limit  ? 
Make  the  statement  using  the  above  symbol. 


194. 


1.  Theorem.  We  now  wish  to  discuss  two  variables, 
each  approaching  its  respective  limit.  If  the  variables  have  a 
constant  ratio ^  what  can  we  say  of  their  limits  f 


2. 


^^^ 


3.  L,et  a  and  ci  be  two  constants,  and  x  and  x  two  vari- 
ables approaching  the  respective  constants  as  limits. 

X 

4.  Also  let  ~,  ^  the  constant,  r. 

X 

5.  \Note  the  variable  x  ==  a.] 

6.  [Also,  note  the  variable  x  ^  a'.] 

X 

7.  Get  a  clear  notion  of  the  ratio,  - ,.     The  ratio  is  cottr 

slant,  although  x  and  xf  are  variables. 

8.  Those  who  fail  to  get  a  clear  conception  of  the  mean- 


132  PLANE  GEOMETRY,  BOOK  III. 

ing  of*  the  above  should  study  the  concrete  ilhistration  given 
below: 


~x ^ 

a  =^a.  constant  (line  6  in.  long). 


a'  =  a.  constant  (line  4  in.  long). 

For  the  purpose  of  illustration  we  take  x,  x\  a,  a\  know?i 
lines.     In  the  theorem  x,  x' ,  a,  a!  are  unknown  quantities;  it 

X 

is  only  known  that   —  =  r,  and  that  r  is  a  constant;  and  we 

X 

.        a         X  a 

zjisk  to  Prove  that  —  =  — , ,  or  — ,  =  y 
^  a         X  a 

Note  variables  x  and  x' . 

(1)  Vi^^  first  take  ;r  =  1^  inches  and  x'  =  1  inch,  we 

have  — ,  =  ~-  z=  -  —  r. 
X  \  ^ 

(2)  If  we  next  take  x  =  ^  inches  and  x'  =-=  2  inches,  we 
have  — ,  =  ty  =^'. 

X  -c 

(3)  If  w^e  next  take  x  =  4:^  inches  and  x  =  S  inches, 

X         4J         3 
we  have  —  ==  -^  =  ^  =r. 

X  o  4i 

Note  — ;  or  r  in  each  case  is  -^  . 
X  ^ 

X 

(4)  No  matter  how  closely  ;r  =  a,  and  x'  =^a\  -7  =  ^  (f )• 

^        a 
In  this  illustration  we  easily  see  that  —  =  — ,    because 

X         a 

a  and  a'  are  known,  being  6  inches  and  4  inches,  respectively. 


rilOPORTION— LIMITS.  133 

9.     In  order  to  grasp  the  full  force  of  this  proof  it  is  abso- 
lutely necessary  for  the  pupil — 

(1)  To  get  a  clear  idea  of  the  meaning  of  each  term  used, 

(2)  And  to  hold  that  idea  firmly  in    mind  during  the 
entire  discussion, 

(3)  And  to  understand  fully  the  authority  for  each  state- 
ment made, 

(4)  And  to  keep  constantly  in  mind  the  steps  whereby 
each  statement  is  established. 

n-    J.  a         X  _ 

10  prove:      — r  r=:  —  —  r. 

^  ax 


Proof:      --=,>^or<r.     [?] 


I.     Prove  —  is  not  greater  than  r.     Suppose,  if  possible , 


that— 

(1)   -,  >n 

[When  the  numerator  is  constant,  the  value  of  a  fraction 
is  decreased  by  increasing  the  denominator.  Then  let  b  rep- 
resent the  required  increase,] 

(3)  Also,  a=^{a!  -^  b)r.     [  ?  ] 

(4)  But  ;^;  =  r  x'\     [  ?  ] 

(5)  And  a  -  xz=^{a:  -  x'  -{-  b)r.     i'>\ 

(6)  Now  a-  x~^',     [  ?  ] 

.  • .  a  —  x  may  be  made  as  small  as  we  choose. 

(7)  And  {a!  -  x')  r  ^  b  r  =  {a!  -  x'  -^  b)  r.     [  ?  ] 

(8)  b  r  —  2l  constant  positive  quantity.     [  ?  ] 

(9)  {a!  —  x)  r  ^=  an  indefinitely  small  but  positive  quan- 
tity.     [?] 

(10)  Then  b  r  >  a  -  x.     [  ?  ] 

(11)  '&yx\,{ci —  x' ^  b)r->br',     [?] 


134  PLANE  GEOMETRY,  BOOK  III. 

(12)  .',{a!  —  x'  ^-  b)r'>a  —  x.     [?] 

But,  above  (in  No.  5)  a  —  ^  =  (a'  —  x  -\-  b)  r,  which  is 
absurd. 

[N.  B.     (5)  is  reached  by  supposing  —    >    r,  but   (12)  is 

based  upon  (6),  (7),  (8),  (9),  (10),  and  (11),  all  of  which  are 
proved  by  Geometry.] 

(13)  Hence  —  is  not  greater  than  r.     [  ?  ] 

a 
II.     Prove  ~r  is  not  less  than  r. 
a 

(1)  Suppose,  if  possible,  \h2X  ~f    <  r. 

(2)  Then  ^^-^  =  r,     [  ?  ] 

a  —  b 

(3)  And  a  =  (a'  —  ^)  r.     [  ?  ] 

(4)  But  Ji;  =  y  r;     [  ?  ] 

(5)  .' .  a  —  x—{a'   -  x'  —  b)  r.     [  ?  ] 

(6)  But  a  —  ;r  is  positive,     [  ?  ] 

(7.)     And  {a'  —  x'  —  b)  r  is  negative;     [  .?  ] 
8)     .  • .  No.  5  is  absurd,     [  ?  ] 

(9)     And  —  is  not  less  than  r.     [  ?  ] 

[The  final  reasoning  is  left  to  the  pupil.] 

195. 

Cor.  I.     If,  while  approaching  their  respective  limits,  two 
variables  are  always  equal,  are  their  limits  equal  ? 
Give  the  proof  of  your  answer. 

196. 

Cor.  II.  If,  while  approaching  their  respective  limits, 
two  variables  have  a  constant  ratio,  and  one  of  them  is  always 
greater  than  the  other,  what  do  you  conclude  about  their 
respective  limits  ? 

Prove  your  answer. 


PROPORTION— LIMITS.  135 


Proportionai,  Lines. 


1.  When  a  line  drawn  parallel  to  the  base  of  a  A  bisects 
one  side,  how  will  it  meet  the  other  side? 

2.  When  a  line  is  drawn  parallel  to  one  of  the  parallel 
sides  of  a  trapezoid  and  bisects  one  of  the  non-parallel  sides, 
how  does  it  meet  the  other  non-parallel  side? 

3.  When  parallel  lines  cut  off  equal  parts  on  a  given 
transversal,  how  will  they  meet  any  other  transversal? 

4.  Draw  a.  scalene  triangle,  ABC,  and  a  line  m  n  \  to 
A  B  cutting  off  J  of  A  C;  what  part  of  B  C  is  cut  cfi7 

197. 

Proposition  XIV. 


Let  A  C  B  be  any  A  and  M  any  point  in  A  C,  and 
M  N  II  to  A  B.  Compare  the  ratio  of  the  parts  of 
A  C  {A  M  :  M  C)  with  the  ratio  of  the  parts  of 
BC{B  N:  N  C). 

Case  I.  Suppose  that  C  M  and  M  A  are  commeiisurable. 
[Review  §  172.]    Apply  a  common  measure,  >^,  to  C  M  and  M  A. 

Then  AM  —pk,  and  M  C  =  ^  >^.     [?] 

Through  points  of  division  of  A  M  and  M  C  draw  lines 
parallel  to  A  B. 


136  PLANE  GEOMETRY,  BOOK  III. 

Then  parts  intercepted  on  B  C  are  equal,  [?]  and  B  N  = 
^/andNC=:^/.     [?] 
[Whsit  is  /?] 

AM  _  B_N 

•*•  M  C  ~  N  C* 

Case  II,     Suppose  A  M  and  M  C  are  incommensurable. 

Divide  C  M  into  any  number,  say  5,  equal  parts.  Call 
each  part  v.  Apply  z^  to  M  A;  it  is  contained  in  M  A  a  given 
number  of  times,  say  /,  with  a  remainder,  7,  which  is  less 
than  V.  lyctter  the  point  at  the  end  of  the  last  measurement 
D,  and  through  D  draw  a  line  ||  to  A  B,  intersecting  C  B  at  E. 


(1)    Then  C  M  :  M  D  =  C  N  :  N  E.     [?] 

Now  V  may  be  made  very  small,  consequently  r  will  ap- 
proach 0  as  a  limit.  But  no  matter  how  small  v  becomes,  (1), 
above,  holds  true. 

What  does  M  D  approach  as  a  limit? 

What  does  N  K  approach  as  a  limit? 

What  does  the  ratio  C  M  :  M  D  approach  as  a  limit? 

What  does  the  ratio  C  N  :  N  B  approach  as  a  limit? 

Why  then  does  CM:MA  =  CN:NB? 

Write  and  carefully  prove  Prop.  XIV. 

i98. 

Cor,  Prove  that  a  line  drawn  ||  to  the  base  through  any 
point  in  the  side  of  the  A  divides  the  other  side  so  that  a  side 


I'ROPORTION— LIMITS.  137 

is  to  either  segment  as  the  other  side  is  to  its  corresponding 
segment. 

199. 
Proposition  XV. 
What  is  the  converse  of  Prop.  XIV.? 


CM      C  N 

Given  A  A  C  B  and  points  M  and  N  so  that;!-^^  =-— — . 

^  MA      N  B 

Can  you  prove  M  N  ||  to  A  B? 

If  not,  draw  M  D  ||  to  A  B. 

What  proportions  follow?  Compare  with  proportion 
given  by  Hyp.  and  show  absurdity. 

{Hi7it.—^Yy  to  show  that  C  D  =  C  N.  Consult  §  192  if 
you  fail.] 

State  and  prove  Prop.  XV.  ^ 

200. 
Proposition  XVI. 

1.  Draw  a  scalene  A  having  sides  4,  5,  and  6.  Biseci 
each  Z  in  turn,  and  each  time  compare  the  ratio  of  the  sides 
of  the  A  including  the  angle  bisected  with  the  ratio  of  the 
segments  of  the  opposite  side  when,  if  necessary,  the  bisector 
is  produced  to  cut  it. 

2.  Draw  other  scalene  As  and  make  similar  comparisons. 
[Hint. — Much  care  must  be  taken  in  the  above  drawings 

in  order  to  reach  any  definite  conclusion  by  the  comparisons.] 


138  PLANE  GEOMETRY,  BOOK  III. 

Try  to  prove  that  the  principle  you  have  discovered  is 
geometrically  true,     li you  fail,  consult  the  hints  given  below. 

\Hints. — Produce  either  side  of  the  Z  bisected  and  through 
the  extremity  of  the  other  side  draw  a  line  parallel  to  the  bi- 
sector. Produce  these  new  lines  until  they  meet,  forming  a 
new  A  wholly  exterior  to  the  original  A-  Then  prove  this 
new  A  is  isos.,  and  then  use  Prop.  XIV.] 

EXKRCISES. 

138.  The  sides  of  a  A  are  5  and  9  feet.  A  line  drawn  || 
to  the  base  divides  the  one  into  segments  of  IJ  and  3 J  feet; 
calculate  the  segments  of  the  other  side. 

189.  The  sides  of  a  A  are  6  and  4  feet,  and  the  base  is 
10  feet.  (1)  Calculate  the  segments  of  the  base  made  by  the 
bisector  of  the  vertical  angle.  (2)  Made  by  the  bisector  of 
each  of  the  other  Zs  when  the  side  opposite  is  considered 
the  base  of  the  A- 

140.  A  line  drawn  i|  to  one  of  the  parallel  sides  of  a  trap- 
ezoid cuts  the  non-parallel  sides  proportionally,  and  each  non- 
parallel  side  is  to  each  of  its  segments  as  the  other  non-parallel 
side  is  to  its  corresponding  segment. 

141.  When  three  or  more  parallel  lines  are  cut  by  two 
transversals,  the  segments  of  either  transversal  are  propor- 
tional to  corresponding  segments  of  the  other  transversal,  and 
either  transversal  included  by  the  parallels  is  to  each  of  its 
segments  as  the  other  transversal  included  by  the  parallels  is 
to  its  corresponding  segment:  and  if  these  transversals  meet 
when  produced,  any  part  between  the  point  of  intersection 
and  a  parallel  is  to  any  of  its  intercepts  as  the  corresponding 
part  of  the  other  transversal  is  to  its  corresponding  intercept. 


PROPORTION— TRIANGLES.  139 

201. 

Proposition  XVII. 


Draw  any  scalene  A  whose  sides  are  known  to  you,  A  B 
C.  Produce  either  side,  A  C.  Bisect  the  exterior  Z  and 
produce  the  bisector  until  it  meets  the  opposite  side,  A  B  pro- 
duced, at  D. 

Compare  the  ratio  of  A  D  to  B  D  with  that  of  .A  C  to  C  B. 

Do  not  jump  to  conclusions,  but  draw  different  shaped  /\s 
and  produce  each  side  in  order  and  bisect  exterior  angle  and 
make  comparisons. 

What  truth  do  you  discover? 

Try  to  prove  it  by  Geometry.  If  you  fail,  try  again, 
using  all  you  know  about  methods  for  discovering  original 
demonstrations. 

The  hint  below  is  given  for  those  who  finally  fail  after 
repeated  trials. 

[/^/«/.— Through  B  draw  line  ||  to  D  C  and  study  the 
segments  of  the  sides  of  A  A  D  C,  etc.,  etc.] 

202. 

Definitions. 

A  line  may  be  divided  into  segments  in  two  ways: 
(1)  when  the  point  of  division  is  iri  the  line,  (2)  when  the  point 
of  division  is  in  the  Xin^ produced.  The  line  is  said  to  be  divided 
inieryially  in  the  first  cases,  and  externally  in  the  second.  In 
each  case  the  segments  are  the  distances  from  the  point  of 
division  to  the  extremities  of  the  line.     In  Prop.  XVII.  A  B 


14C  PLANE  GEOMETRY,  BOOK  III. 

is  said  to  be  divided  externally  at  D,     What  are  the  segments 
of  A  B  when  divided  externally  at  D  ? 

Draw  a  line  and  divide  it  (1)  internally,  (2)  externally. 
Tell  the  segments  in  each  division. 

Exercises. 

142.  State  and  prove  the  converse  of  Prop.  XVI. 

143.  State  and  prove  the  converse  of  Prop.  XVII. 

203. 

Proposition  XVIII. 

1.  Draw  two  parallels  5  in.  and  3  in.  in  length.  Join  their 
extremities  and  produce  the  lines  until  they  meet.  Through 
this  point  draw  a  transversal  which  cuts  off  one-half  of  the 
longer  parallel.  How  is  the  shorter  parallel  cut?  Draw  a 
transversal  which  cuts  off  one-fourth  the  shorter  parallel. 
How  does  it  cut  the  longer  parallel  ? 

2.  Draw  any  two  parallels  and  draw  three  or  more  trans- 
versals which  meet  at  a  common  point.  Compare  the  inter- 
cepts on  one  parallel  with  the  intercepts  on  the  other  paral- 
lel.    Are  the  i7itercepts  proportional  ? 

State  and  prove  Prop.  XVIII. 


Prove  a'  \  a^=^  b'  '.  b.  Draw  through  I  and  J  lines  parallel 
to  E  H  and  study  As  E  F  H  and  E  G  H.  Also  study  the 
quadrilaterals  formed.] 

204. 

Cor.  In  the  above  figure  prove  a!  :  ^=FE:IE  = 
H  E  :  K  E,  etc. 


PROPORTION— SIMILAR  FIGURES.  )  141 

SIMII.AR  FIGURES. 

205. 

When  are  two  figures  equivalent  ?  equal  ?  May  a  square 
equal  a  triangle  ?  a  rectangle  ? 

(1)  Equal  figures  are  not  only  equal  in  area,  but  they  are 
similar  hi  form;  as  we  have  learned,  they  can  be  made  to  coin, 
cide  in  all  their  parts. 

(2)  Equivalent  figures  are  equal  i?i  area,  but  they  are  not 
necessarily  similar  in  form. 

(3)  Similar  polygons  are  mutually  equiangular,  and  their 
homologous  sides  are  proportional  ? 

Draw  any  A  and  assume  any  side  as  the  base;  then  draw 
a  line  which  is  parallel  to  the  base  and  which  cuts  off  \  of  one 
side.  How  does  it  cut  the  other  side?  Note  the  two  trian- 
gles. Are  they  equiangular?  [  ?  ]  Are  their  sides  propor- 
tional?    [  ?  ]     Are  they  similar?- 

4.  Homologous  parts,  sides,  angles,  etc.,  are  those  which 
are  similarly  situated. 

5.  Si7nilar  arcs  subtend  equal  central  angles;  similar 
sectors  and  segmeyits  are  those  whose  arcs  subtend  equal  cen- 
tral angles. 

Are  all  circles  similar? 

206. 
Proposition  XIX. 

(1)  Draw  any  triangle,  assume  any  side  the  base,  and 
fix  any  point  in  either  of  the  sides.  Let  the  A  be  A  B  C,  and 
A  B  the  base,  and  P  the  point  fixed  in  A  C.  Now  draw  a  line 
through  P  which  cuts  B  C  at  D  and  makes  the  Z  C  P  D  = 
Z  C  A  B.  Is  P  D  parallel  to  the  base?  Can  j^ou  prove  the 
triangles  similar? 

(2)  Given  the  triangles  ABC  and  D  E  F  in  which  angles 
A,  B,  and  C  are  respectively  equal  to  angles  D,  E,  and  F. 
Can  you  prove  the  triangles  similar? 


142  PLANE  GEOMETRY,  BOOK  III. 

Can  you  prove  two  triangles  similar  which  are 
mutually  equiangular? 

[See  (2)  above.] 

What  two  properties  have  similar  polygons?    [See  §  205,  3.] 

[^Hint. — Superpose.] 

If  y^z^ /a27,  after  using  the  above  exercises  and  hints  as 
helps,  see  the  further  hint  given  below. 

{^Hint. — Superpose,  placing  an  Z  upon  its  equal  Z  and 
similar  side  upon  similar  side.  Show  that  sides  are  parallel; 
.  • .  sides  are  proportional.] 

207. 

Cor,  I.  Given  any  two  triangles  in  which  two 
angles  of  one  equal  two  angles  of  the  other.  Prove 
the  triangles  similar. 

208. 

Cor,  II.  Given  two  rt.  triangles  in  which  an 
acute  Z  of  one  equals  an  acute  angle  .of  the  other. 
Prove  the  triangles  similar. 

209. 

Cor.  J 11  Given  two  triangles  similar  to  a  third 
triangle.     Make  deduction  and  prove  it. 

Exercises. 

144.  Given  two  unequal  lines,  a  and  b.  Construct  two 
squares  having  these  lines  respectively  for  sides.  Can  you 
prove  these  squares  similar?     Make  general  conclusion. 

145.  Given  two  equilateral  triangles  having  unequal 
sides      Can  you  prove  them  similar? 

146.  Given  two  isosceles  As  having  vertical  angles  equal. 
Can  you  prove  them  similar  ? 


PROPORTION— SIMILAR  FIGURES.  143 

147.  Given  two  regular  hexagons.  Prove  that  they  are 
similar. 

148.  Given  two  regular  polygons,  each  having  n  sides. 
Prove  that  they  are  similar. 

210. 

Proposition  XX. 

You  have  proved  in  Prop.  XIX.  that  if  two  triangles  are 
mutually  equiangular,  the  corresponding  sides  are  propor- 
tional, and  consequently  the  triangles  are  similar. 

Now  can  you  prove  that  two  triangles  are  simi- 
lar if  the  sides  of  one  are  proportional  to  the  sides  of 
the  other ^  each  to  each  f 

This  proof  is  difficult,  so  do  not  get  discouraged  if  you 
fail.  But  make  a  stubborn  fight  before  you  give  it  up.  Re- 
view all  the  suggestions  given  to  pupils  for  attacking  original 
demonstrations. 

If,  having  exhausted  all  your  powers,  you  fail,  consult  the 
"hint,"  prove  the  Prop,  yourself  without  further  reading  just 
as  soon  as  you  discover  the  proof,  and  compare  your  proof 
with  the  rest  of  the  "hint."  Possibly,  part  of  your  proof  will 
be  original.  Try  to  see  just  what  effort  you  failed  to  put  forth 
in  your  trying  to  discover  the  demonstration.  This  may  give 
success  to  future  efforts. 

[//z«/.— Given  the  two  triangles  ABC  and  D  E  F,  in 
which  AB  :DE==BC:EF=AC:DF.  (Pupil  may 
draw  the  As  having  sides  4",  5",  and  6",  and  6",  7^",  and  9", 
respectively.  Make  the  sides  of  A  B  C  longer  than  those  of 
D  E  F.)  On  C  A  measure  off  C  G  =  F  D  and  on  C  B  meas- 
ure off  C  H  =  F  E.  Draw  G  H.  (Now  try  to  prove  As 
C  G  H  and  CAB  similar.  Then  try  to  prove  /\s  C  G  H  and 
D  E  F  equal.) 


I44  PLANE  GEOMETRY,  BOOK  III. 

(1)  AC  :  D  F=  BC:  E  F.     [?  ] 

(2)  Then  AC:GC=BC:HC      [?] 

(3)  As  A  B  C  and  G  H  C  are  similar.     [  ?  ] 

(4)  A  C  :  G  C  =1^  A  B  :  G  H.     [  ?  ] 

(5)  A  C  :  D  F  =:^  A  B  :  D  K.     [  ?  ] 

(6)  A  C  :  G  C  =  A  B  :  D  E.     [  ?  ] 
(Compare  (4)  and  (6)  and  make  deductions.) 

(7)  .  • .  G  H  =  D  E.     [  ?  ] 

(8)  And  A  s  G  C  H  and  D  E  F  are  equal.     [  ?  ] 

(9)  Also  As  A  B  C  and  D  E  F  are  similar,  Q.  E.  D.] 
We  have  seen  that  if  triangles  are  equiangular,  their  sides 

must  be  proportional,  and  conversely.  Would  you  form  the 
same  conclusion  about  equiangular  quadrilaterals  and  other 
equiangular  polygons? 

(1)  Compare  a  square  with  a  rectangle,  a  rhombus  with 
a  rhomboid.     (Draw  figures.) 

(2)  Also  compare  a  square  with  a  rhombus,  a  rectangle 
with  a  rhomboid.     (Their  sides  may  be  equal  or  proportional.) 

(3)  What  conclusion  do  you  reach  ? 

211. 
Proposition  XXI. 

Draw  two  As  A  B  C  and  D  E  F,  making  A  B  =  10", 
A  C  =  8",  B  C  =  6'',  and  D  E  ==  5",  D  F  =  4",  E  F  ^  3'. 
Are  the  As  similar?     Why  ? 

Draw  the  altitudes  upon  A  B  and  D  E.  Compare  them. 
Do  they  have  the  same  ratio  as  A  B  and  D  E?  as  A  C  and 
D  F?  as  BCand  E  F? 

Draw  the  other  altitudes  and  compare  their  ratio  with 
that  of  any  two  homologous  sides. 

Draw  other  similar  triangles  and  make  further 
comparisons  of  the  ratio  of  similar  altitudes  to  the 
ratio  of  homologous  sides. 


PROPORTION— SIMILAR  FIGURES.  145 

What  deduction  can  you  make  of  the  altitudes  of 
all  similar  triangles  ? 

Write  Prop.  XXI.  and  its  formal  proof. 

What  is  the  converse  of  Prop.  XXI  ?  Can  you  prove  it  ? 

Exercises. 

149.  In  a  city  are  two  rectangular  lots,  one  is  50  feet  by 
15C  feet,  and  the  other  is  100  feet  by  250  feet.  Are  they  sim- 
ilar?   Why? 

150,  Draw  any  scalene  triangle.  Can  you  draw  an 
isosceles  triangle  similar  to  the  triangle  drawn  ?  Prove  your 
answer. 

212. 

Proposition  XXII. 

Problem,  To  divide  a  line  into  parts  propor- 
tional to  two  or  more  given  lines. 

There  are  two  equal  fractions  whose  numerators  are  3 
and  4,  respectively,  and  the  sum  of  whose  denominators  is  49. 
Required  the  denominators. 

(1)  Solve  by  Algebra. 

(2)  Solve  by  Geometry. 

\^Hint. — Use  lines  to  represent  the  numbers.] 
[Hint. — li  you  fait  on  (2),  draw  any  scalene  A  and  fix  a 
point  in   one  side.     Note  how  the  point  divides  the   side. 
Now,  how  can  you  divide  the  other  side  into  parts  propor- 
:onal  to  these  two  ?] 

There  are  two  equal  fractions,  the  first  is  f  and  the  nu- 
merator of  the  second  is  3.  Required  the  denominator  of  the 
second. 

(1)  Solve  by  Algebra„ 

(2)  Solve  by  Geometry. 


i46  PLANE  GEOMETRY,  BOOK  III. 

213. 

Proposition  XXIII. 

Problem,  To  find  a  fourth  proportional  to  three 
given  lines. 

214. 

Proposition  XXIV. 

(1)  Draw  any  acute  Z ,  cl,  and  any  line,  b.  Then  draw 
two  Z  s,  each  equal  to  Z  «,  but  the  sides  of  one  3  b  and  4  b, 
while  the  sides  of  the  other  are  6  b  and  8  b.  Join  the  ex- 
tremities of  these  sides,  forming  two  As.  Compare  the  re- 
maining Z  s  of  these  As.  Compare  the  remaining  sides  of 
these  As.  What  deduction  can  you  make  respecting  the  two 
triangles? 

(2)  Draw  any  obtuse  Z  ,  c,  and  any  two  unequal,  lines,  d 
and  e.  Then  draw  two  As  each  having  an  Z  =  Z  ^  and  the 
sides  including  Z  ^  of  one  2  d  and  3  e  and  the  sides  including 
/_c  o{  the  other  6  d  and  9  e. 

Make  deductions  respecting  these  As. 

(3)  What  deduction  in  general  ^HciowX,  l\^  X^idX 
have  an  Z  of  one  equal  to  an  I.  of  the  other  and  the 
siaes  including  the  equal  angles  proportional  f 

State  and  prove  Prop.  XXIV. 

Exercises. 

151.  Two  sides  of  a  A  are  8  inches  and  10  inches,  and 
the  base  is  12  inches.  If  a  line  9  inches  long,  parallel  to  the 
base,  terminates  in  the  sides,  what  are  lengths  of  the  segments 
of  the  vsides? 

152.  The  sides  of  a  triangle  are  5, 7,  and  9,  and  the  short- 
est side  of  a  similar  A  is  14.     Find  the  remaining  sides. 

153.  The  base  of  a  triangle  is  10"  and  the  altitude  is  6''. 
In  a  similar  triangle,  the  base  is  unknown  and  the  altitude  is 
15".     What  is  the  base? 


PROPORTION— SIMILAR  FIGURES.  147 

154.  Given  the  scalene  acute-angled  A  having  the  sides 
a,  b,  c,  and  the  altitude  upon  a,  m;  the  altitude  upon  b,  n;  and 
the  altitude  upon  c,  o.     Prove  that 

(1)  a  \  b  ^=^  n  '.  m. 

(2)  a  :  c  ^=^  0  :  fn. 

(3)  b  :  c  =^  o  :  n. 

Write  general  statement  of  the  altitudes  of  a  triangle. 
[If  you  fail  in  proof,  see  §  208.] 

215. 

Proposition  XXV. 

(1)  Draw  two  Zs  having  sides  respectively  parallel  and 
extending  the  same  direction  from  the  vertex.  What  deduc- 
tion can  you  make  respecting  these  angles?     Prove  it. 

[Hint. — Produce  a  side  of  one  until  the  non- correspond- 
ing side,  or  that  side  produced,  is  cut.] 

(2)  Draw  two  Z  s  having  sides  respectively  parallel  and 
extending  respectively  in  opposite  directions  from  the  vertex- 

[Note.— Such  lines  are  said  to  be  aw/z-parallel.  In  (1) 
they  are  said  to  be  SYM-parallel.] 

Can  you  make  the  same  deduction  about  these  Z  s  that 
you  made  in  (1)? 

(3)  Draw  two  Z  s,  a  side  of  one  sym-parallel  to  a  side  of 
the  other,  but  the  other  side  of  the  first  anti-parallel  to  the 
other  side  of  the  second. 

Make  deduction  and  prove  it. 

Make  a  general  statement  respecting  the  three 
deductions  made  and  proved. 

Use  the  fewest  possible  words  necesary  to  make  the  state- 
ment clear  and  comprehensive. 

216. 

Cor.  I.  (I)  Can  you  draw  two  unequal  As  having  all 
pairs  of  corresponding  sides  parallel  and  extending  the  same 


148  PLANE  GEOMETRY,  BOOK  III. 

direction  from  the  vertex,  or  sym-parallel?     If  so,  what  de- 
duction can  you  make  of  these  As? 

(2)  Can  you  draw  two  unequal  /\s  having  all  pairs  of 
corresponding  sides  anti-parallel?  If  so,  what  deduction  can 
you  make? 

(3)  Can  you  draw  two  unequal  As  having  from  the  vertex 
of  each  angle  one  pair  of  sides  sym-parallel  while  the  other 
pair  are  anti-parallel? 

Prove  your  answer. 

What  general  conclusion  do  you  reach  about  two 
As  whose  corresponding  sides  are  parallel? 

[Note  that  parallel  lines  may  be  either  sym-parallel  or 
anti-parallel.] 

Sjate  and  carefully  prove  Cor.  I.,  Prop.  XXV. 

217. 
Proposition  XXVI. 

(1)  Draw  two  acute  Zs  having  the  sides  of  one  perpen- 
dicular to  the  sides  of  the  other.    Make  deduction  and  prove  it. 

(2)  Draw  two  obtuse  Z  s  having  the  sides  of  one  per- 
pendicular to  the  sides  of  the  other.  Make  deduction  and 
prove. 

(3)  Draw  an  acute  angle  and  an  obtuse  angle  having 
the  sides  of  one  perpendicular  to  the  sides  of  the  other.  Com- 
pare the  angles  and  prove  deduction  made. 

(4)  What  general  deduction  can  you  make 
about  any  two  Z  s  which  have  their  sides  respectively 
perpendicular? 

State  and  prove  Prop.  XXVI. 

218. 

Cor.  I.  (1)  Draw  two  acute-angled  triangles  having  the 
sides  of  one  perpendicular  to  the  sides  of  the  other.     Com- 


PROPORTION— SIMILAR  FIGURES.  149 

pare  the  angles  included  by  the  sides  which  are  respectively 
perpendicular  to  each  other. 

(2)  Draw  two  obtuse-atigled  triangles  having 
sides  respectively  perpendicular.  Make  deductions 
and  prove  them. 

Make  further  drawings  in  which  triangles  have 
their  sides  respectively  perpendicular. 

State  Cor.  I.,  Prop.  XXVI.,  and  prove  it. 
Be    sure    that    your    proof    comprehends    all    possible 
conditions. 

219. 
Proposition  XXVII. 

Draw  a  rt.  A  and  consider  the  hypotenuse  the  base. 
Draw  the  altitude  and  compare  — 

(1)  The  original  rt.  A  with  each  A  formed  by 
the  altitude  ; 

(2)  The  rt.  As  formed  by  the  altitude; 

(3)  The  altitude  with  the  segments  of  the 
hypotenuse; 

(4)  Each  leg  with  the  whole  hypotenuse  and 
the  adjacent  segment. 

State  Prop.  XXVII.,  which  should  comprehend  the  four 
deductions  made. 

220. 

Cor. 7,  Prove  that  the  square  of  the  altitude 
upon  the  hypotenuse  of  a  rt.  A  equals  the  product 
of  the  segments  of  the  base. 


150  PLANE  GEOMETRY,  BOOK  III. 

221. 

Cor.  II.  When  the  altitude  of  a  rt  A  upon  the 
hypotenuse  is  drawn,  the  square  of  either  leg  is  equal 

[Pupil  will  complete  the  sentence  and  prove.] 

222. 

Cor.  Ill  The  squares  of  the  legs  of  a  rt.  A  are 
proportional  to  the  adjacent  segments  made  by  the 
altitude  drawn  to  the  hypotenuse. 

[//^;^A-Use§22L] 

223. 

Cor.  IV.  Draw  a  semicircle  and  fix  any  point 
in  the  circumference.  Draw  a  perpendicular  from 
the  fixed  point  to  the  diameter.  (1)  Compare  the 
perpendicular  with  the  segments  of  the  diameter. 
(2)  Draw  chords  from  the  fixed  point  to  the  ends 
of  the  diameter.  Compare  each  chord  with  the  diam- 
eter and  the  adjacent  segment. 

State  Cor  IV.,  Prop  XXVII. 

224. 

Proposition  XXVIII. 
Problem.      To  construct  a  mean  proportional  to 
two  given  straight  lines. 

KxKRCISKS. 

155.  Construct  a  square  equivalent  to  a  given  rectangle 
(assuming  that  the  area  of  a  rectangle  equals  the  product  of 
the  base  by  the  altitude). 


PROPORTION— SIMILAR  FIGURES.  151 

156.  Construct  a  square  equivalent  to  a  given  rt.  A- 

157.  Given  the  side  of  a  square  and  one  side  of  an  equiv- 
alent rectangle.     Construct  the  square  and  the  rectangle. 

Consult  the  hint  if  j/^w /a//.  It  is  not  difi&cult.  Do  not 
give  up  easily, 

\^Hint. — I.  Can  you  find  a  third  pro portiojial  to  two  given 
lines?  or,  II.  Construct  Ex.  156  again  and  try  to  determine 
how  you  could  replace  one  segment  of  the  hypotenuse  if  it  and 
the  semicircumference  were  erased.  If  you  still  fail,  construct 
the  entire  circle  in  the  above;  produce  the  altitude  below  the 
diameter  until  the  lower  semicircumference  is  cut.  How 
many  points  on  the  circumference  do  you  now  have?  How 
many  of  these  points  would  be  erased  if  the  required  segment 
of  the  diameter  (required  side  of  the  rectangle)  were  erased? 
How  many  points  on  the  circumference  are  required  to  deter- 
mine the  center?] 

225. 

Proposition  XXIX. 
Carefully    draw    the    figure    and    review  Section    221. 
What  does  the  square  of  each  leg  equal?     What  does  the  sum 
of  the  squares  equal? 

Compare  the  sum  of  the  squares  with  the  square  of 
the  hypotenuse. 

State  Prop.  XXIX. 

226. 

Cor.  I.  What  does  the  square  of  either  leg  equal  when 
compared  with  the  square  of  the  hypotenuse  and  the  square 
of  the  other  leg? 

Exercises. 

158.  Construct  a  square  having  twice  the  area  of  a  given 
square. 

159.  Construct  a  square  having  the  area  (1)  of  three  un- 
equal given  squares,  (2)  of  three  equal  given  squares. 


152  PLANE  GEOMETRY,  BOOK  III. 

160.  Given  3  unequal  lines,  a,  b,  and  c.  Construct  a 
square  equivalent  to  2  a  2  _[_^  2  —  ^2^  jg  ^[jjs  ever  impos- 
sible ? 

161.  Draw  an  indefinite  straight  line  and  cut  off  equal  dis- 
tances. Using  these  parts,  construct  (1)  \l'l,  (2)  v^8,  (3)  n/5 
(two  ways),  (4)  \/6,  (5)  v'H  (two  ways),  (6)  v/29  (two  ways), 
(7)   \/24  (two  ways). 

l^Hint. — Make  a  list  of  perfect  squares.  Find  sums,  dif- 
ference, etc.] 

162.  What  is  the  length  of  the  tangent  to  a  circle  whose 
diameter  is  14,  from  a  point  whose  distance  from  the  center 
is  25. 

163.  Lay  off  equal  parts  on  an  indefinite  line  and  con- 
struct v'3.  Calculate  the  altitude  of  an  equilateral  A  whose 
base  is  4  v^8. 

Construct  the  A  and  compare  its  altitude  with  result  ob- 
tained by  calculation. 

227. 
Proposition  XXX. 

Given  two  polygons  which  are  composed  of 
the  same  number  of  triangles  which  are  similar 
each  to  each  and  are  similarly  placed.  Compare  the 
polygons. 

What  are  similar  polygons?  Make  deduction  and 
prove  it. 


PROPORTION— SIMILAR  FIGURES. 


153 


{^Hint. 


Given  the  polygons  A  B  —  E  and  F  G  —  J,  composed 
of  the  similar  As,  which  are  similarly  placed. 

(1)  ABC  and  F  G  H. 

(2)  A  C  D  and  F  H  I. 

(3)  A  D  E  and  F  I  J. 

To  prove  the  polygons  similar. 

Proof: 

Z   1  -   Z    2;      [  ?  ] 

Z    3  r=    Z     4;      [  ?  ] 

.-.  Z  C=  Z    H.     [?] 

In  like  manner  we  can  prove  other  corresponding  Z  s  of 
the  polygons  equal. 

[?] 
[?] 
[?] 


A 

C        C    B 

F 

H  ~  H  G' 

A 

C        CD 

F 

H        H    I 

, 

C   B  _  C 

D 


HG 


H    I* 


154  PLANE  GEOMETRY,  BOOK  III. 

In  like  manner  the  remaining  corresponding  sides  may 
be  proved  proportional. 

.  • .  the  polygons  are  similar.     [  ?  ]  ] 

228. 

Proposition  XXXI. 

State  and  prove  the  converse  of  Prop.  XXX. 

See  "hint,"  if  you  fail. 

S^Hint. — Given  two  similar  polygons.  Can  you  prove 
them  composed  of  the  same  number  of  A^,  similar  each  to 
each  and  similarly  placed  ? 

(RedraviT  the  figure  of  Prop.  XXX.) 

(1)     Z  B  =  Z  J;     [  ?  ] 

(3)  .  • .  As  A  E  D  and  F  J  I  are  similar.     [  ?  ] 

.     (4)  Z  D  =  Z   X.     [  ?  ] 

(5)  Z7=  Z8.     [?] 

(6)  Z5=  Z6.     [?] 


(7) 

(8) 


ED        DA 


J  I  IF       "-  ' 

ED  _  D^ 
JI   ~  I  H'     L-" 
Pupil  will  finish  the  proof.] 


229. 

Proposition  XXXII. 

Given  two  similar  polygons.  Compare  the  ratio  of  their 
perimeters  with  the  ratio  of  any  pair  of  homologous  sides. 

\^Hint. — Draw  similar  polygons  having  sides,  a,  b,  c,  d,  etc., 
and  «',  b\  c\  d\  etc.     Then  we  are  required  to  prove  that 

Tf   —   T'  —  "7  —  ~Jf  —    ••••J 


a'  -\-  b'  ^  c'  ^  d'  ^   :...         a'         b'        c'        d' 


proportion—similar  figures.  155 

Exercise. 

164.  The  perimeters  of  two  similar  polygons  are  119  and 
68;  if  side  of  the  first  is  21,  what  is  the  homologous  side  of 
the  second? 

230. 

Definition.  (1)  The  projection  of  a  point  upon  a  straight 
line  is  the  foot  of  the  perpendicular  drawn  from  the  point  to 
the  line. 

(2)  The  projection  of  a  finite  straight  line  upon  a  line  is 
that  portion  of  the  second  line  included  between  the  projec- 
tions of  the  ends  of  the  finite  line. 

Exercises. 

165.     I.  Draw  a  scalene  A  and  assume  the  longest  side  the 
base. 

(1)  What  is  the  projection  of  the  vertex  upon  the 
base? 

(2)  Show  the  projection  of  each  side  upon  the  base. 
II.  Draw  a  rt.  A  and  assume  either  leg  the  base. 

(1)  What  is  the  projection  of  the  hypotenuse  upon 
the  base? 

(2)  What  is  the  projection  of  the  vertex? 

(3)  What  is  the  projection  of  the  other  leg? 

III.  Draw  any  obtuse  scalene  A  and  assume  the  short- 
est of  the  two  sides  including  the  obtuse  Z  the 
base. 

(1)  What  is  the  projection  of  the  vertex  of  the  A 
upon  the  base,  or  the  base  produced? 

(2)  What  is  the  projection  of  the  side  opposite  the 
obtuse  Z? 

(3)  What  is  the  projection  of  the  other  side  includ- 
ing the  obtuse  Z  ? 

Compare  these  projections  with  those  when  each  of  the 
other  sides  of  the  A  i^  assumed  the  base. 


156 


PLANE  GEOMETRY,  BOOK  III. 


166.     (1)     Can  the  projection  of  a  line  ever  be  longer  than 
that  line? 

(2)  When  does  the  projection  of  a  line  equal  that 
line? 

(3)  When  is  the  projection  of  a  line  shorter  than 
the  line?     When  is  it  a  point. 

231. 

Proposition  XXXIIT. 


1.  In  the  acute-angled  ^^  A  B  C  what  is  the  projection 
ofBC?  of  AC?  ofC? 

2.  It  is  desired  to  find  the  relation  the  square  of  a  side 
opposite  an  acute  Z  (in  the  /^  A  B  C  al/  the  Z  s  are  acute)  to 
the  squares  of  the  other  sides  and  the  rectangle  formed  by 
one  of  them  and  the  projection  of  the  other  side  upon  it. 

If  we  take  Z  A,  we  wish  to  find  the  relation  of  d^  to  a^, 
r ,  and  rectangle  formed  by  c  and  d. 

(1)  ^^r=m^+;^^;       [?] 

(2)  m^  =  «^  -  ^^     L  ?  ] 


(3) 


d)'' 


^  d'-2cd',     [  ?  ] 


n'  :=  {c 

.-.     d^=}     [?] 

State  the  deduction  proved. 

Let  the  pupil  find  the  value  of  a^  [opposite  Z  B]  in  terms 
d,  c,  and  n;  also  the  value  of  <:^. 

3.  In  the  obtuse-angled  A  D  E  F  find  the  value  of  s^, 
or  of  t\ 

(I)  0/s'  in  terms  of  ^^  /^  and  oL  s'  =  r'  -^  p\  [  ?  ] 
Then  find  the  value  of  r^  and  p^  and  substitute. 


PROPORTION— SIMILAR  PIGTJRES.  157 


(2)     Find  the  value  of  /\ 

Make  general  deduction  about  the  square  of  the  side  oppo- 
site an  acute  angle  in  any  A- 

232. 

Proposition  XXXIV. 

What  does  the  square  of  any  side  opposite  an 
obtuse  Z  in  any  A  equal  ? 

[See  A  D  E  F,  §  281.] 

[What  is  the  projection  of  s  upon  /  (produced)  ?] 

Discuss  all  possible  cases  and  form  general  conclusion. 

233. 

Proposition  XXXV. 

Draw  any  A  and  assume  any  side  the  base  and  draw  the 
median  from  the  vertex  to  the  base. 

1.  Compare  the  sum  of  the  squares  of  the  other  two 
sides  of  the  A  with  the  square  of  the  median  and  the 
square  of  half  the  base. 

(What  does  the  sum  of  the  squares  of  the  sides  equal  in 
terms  of  median  and  half  the  base?) 

2.  Compare  the  difference  of  the  squares  of  the  other  two 
sides  with  the  rectangle  formed  by  the  base  and  the  pro- 
jection of  the  median  upon  the  base. 

State  the  general  conclusion  comprehending  the  results 
of  both  of  the  above  comparisons. 


(1)  a=  =  (t  +  oY  +  q\     [  ?  ] 

(2)  b'=p'-\-g\  or 

(2)  b' =  (n-oY  +  q-';     [?] 

(3)  .-.a'  +  b'  =  2n''  +  2m\     [  ?  ]     Or, 

(4)  a'  —  b'  =itio  =  %so.     [  ?  ]     ] 


158  PLANE  GEOMETRY,  BOOK  III.  ' 

234. 

Proposition  XXXVI. 

Fix  any  point  in  a  circle  and  through  it  draw  any.  two 
chords. 

Compare  the  product  of  the  segments  of  one 
chord  with  the  product  of  the  segments  of  the  other. 
What  conclusion  do  3^ou  reach  .^ 

Write  and  prove  Prop.  XXXVI. 

[f/inL— Form  two  As  by  joining  the  ends  of  the  chords. 
Compare  the  A^-] 

235. 
Proposition  XXXVII. 


Given:     A  secant,  A  B,  and  a  tangent,  A  C,  drawn  from 
a  point  without  the  circle. 

Compare   the   square  of  the  tangent   with   the 
product  of  the  secant  and  its  external  segment. 

State  and  prove  Prop.  XXXVII. 

'[Hint. — Note  the  As,  which  are  similar?     What  propor- 
tion  can  you  form  using  A  B,  A  D,  and  AC?] 


PROPORTION— SIMILAR  FIGURES.  159 

236. 

Cor .  I.  The  tangent  is  a  mean  proportional 
between .... 

Finish  and  prove. 

237. 

Cor,  11.  Draw  another  secant  and  compare 
the  products  of  each  whole  secant  and  its  external 
segment. 

Exercises. 

167.  Construct  a  fourth  proportional  to  3  given  lines, 
using  §  237. 

168.  The  length  of  the  common  chord  of  two  intersect- 
ing circles  is  16,  and  their  radii  are  10  and  17.  What  is  the 
distance  between  their  centers  ? 

169.  C  and  D  are  the  middle  points  of  a  chord,  A  B,  and 
and  its  subtended  arc.  If  A  D  =  13  and  CD"  12,  what  is 
the  diameter  of  the  circle  ? 

170.  In  a  scalene  A  two  sides  are  8"  and  5"  and  the  pro- 
jection of  the  median  drawn  to  the  third  side  upon  that  side 
is  3". 

Required  the  third  side. 

171.  The  three  sides  of  a  A  are  x,  y,  and  z.     1^  x  ^=  8, 

7 
jK  =  6,  and  the  projection  of  y  upon  ^  =  ^  yT'^   calculate 

2,  2  being  opposite  an  acute  Z . 

172.  The  radius  of  a  circle  is  8",  and  a  tangent  to  the 
circle  is  15".  What  is  the  length  of  a  secant  drawn  from  the 
same  point  as  the  tangent,  (1)  if  the  secant  passes  through  the 
center,  (2)  if  the  distance  from  the  center  to  the  secant  is  5"  ? 

173.  A  side  of  a  rhombus  is  10  and  one  diagonal  is  52. 
Find  the  other  diagonal. 


160  PLANE  GEOMETRY,  BOOK  III. 

238. 

Proposition  XXXVIII. 
Draw  any  scalene   triangle  and  bisect  the  vertical  angle; 
produce  the  bisector  to  meet  the  base. 

Compare  the  rectangle  formed  by  tlie  two  sides 
of  the  A  with  the  rectangle  made  by  the  segments 
of  the  base  plus  the  square  of  the  bisector. 

[^Hmt. — Circumscribe  a  circle  about  the  A-  Produce  the 
bisector  until  it  becomes  a  chord  of  the  circle.  Join  the  end 
found  to  that  vertex  of  the  A  niade  by  the  base  and  the 
shortest  side.  Make  deductions.  What  equal  Z  s  do  you  find? 
What  similar  A^?  What  proportions  involving  sides  of  simi- 
lar As?  involving  interisecting  chords?] 

Exercises. 

174.  The  base  of  a  A  is  15",  and  the  sides  are  7'  and 
10'',  and  the  length  of  the  bisector  of  the  vertical  Z  is  4".  Find 
the  segments  of  the  base. 

175.  The  base  of  a  A  is  20",  and  the  sides  are  12"  and 
14".  Calculate  the  bisector  of  the  vertical  angle  when  the 
shortest  segment  is  6". 

239. 

Proposition  XXXIX. 

Draw  any   /\  and  its  circumscribing  circle.     Draw  the 

altitude  to  the  longest  side  as  base  and  draw  the  diameter 

through  the  vertex  of  the  A-     Also  join  the  other  end  of  the 

diameter  to  the  other  end  of  the  shortest  side. 

Compare  the  rectangle  of  the  sides  of  the  Z\  with 
the  rectangle  of  the  diameter  and  the  altitude. 

\_Hi7it. — How  many  rt.  As  in  the  figure?  Can  you  dis- 
cover any  similar  rt.  As?] 

Draw  the  altitude  to  a  shorter  side  and  prove  that  the 
above  deduction  is  true. 


PROPORTION— SIMILAR  FIGURES.  161 

Exercises. 

176.  The  sides  of  a  A  are  12,  18,  and  24  units,  respect- 
ively.    Find  the  diameter  of  the  circumscribed  circle. 

Test  the  accuracy  of  your  answer  by  a  drawing. 

177.  Given  the  sides  of  the  following  As.  Is  the  great- 
est angle  of  each  acute,  right,  or  obtuse?  Give  full  reason  for 
each  answer. 

(1)  3,  5,  and  6. 

(2)  3,  4,  and  5. 

(8)     8,  9.  and  12.  " 

(4)  16,  17,  and  25. 

(5)  10,  11,  and  16. 

178.  (1)  Can  you  show  that  the  diagonals  of  a  trape- 
zium divide  it  into  four  As  which  are  proportional? 

(2)  Draw  three  equal  circles,  tangent  to  each  other,  and 
join  their  centers.  Compare  this  A  with  the  one  formed  by 
joining  the  points  of  contact. 

179.  A  carpenter  wishes  to  brace  a  corner  post;  the 
piece  of  timber  he  proposes  to  use  is  7  feet  long.  If  he  sets 
the  brace  4  J  feet  from  the  foot  of  the  corner  post,  how  high 
will  it  reach  ? 

180.  What  is  the  longest  line  that  can  be  drawn  in  a 
room  20  feet  by  30  feet  by  10  feet? 

181.  A  stair-builder  wishes  to  cut  a  diagonal  brace  for  a 
flight  whose  horses  are  3  feet  apart.  If  the  horizontal  dis- 
tance between  the  extremities  of  the  stairs  is  8  feet,  and  the 
vertical  distance  7  feet,  how  Icng  must  the  brace  be? 

182.  Produce  two  equal  chords  till  they  meet.  What 
can  you  prove  concerning  the  secants  and  their  external 
segments  ? 

183.  What  is  the  length  of  a  rafter  which  projects  over 
the  outside  of  the  plate  15  inches,  if  the  comb  of  the  house  is 
8  feet  above  the  level  of  the  plate  and  the  house  is  24  feet 
wide? 

184.  If  two  secants  drawn  from  the  same  point  without 
u — 


162  PLANE  GEOMETRY,  BOOK  III. 

the  circle  are  equally  distant  from  the  center,  how  are  they 
related  ? 

185.  Given  any  two  intersecting  circles  and  their  com- 
mon chord  produced.     From  any  point  on  this  chord  produced 

draw  a  tangent  to  each  circle  and  compare  them. 

SUPPIvEMENTARY   KxERCISKS. 

186.  If  one  leg  of  a  right  triangle  is  double  the  other, 
show  how  the  perpendicular  from  the  vertex  of  the  right 
angle  to  the  hypotenuse  divides  it. 

187.  T  and  W  are  the  mid  points  of  a"  chord  R  S  and  its 
subtended  arc  respectively.  If  R  W  ==  9,  and  T  W  =  3,  what 
is  the  diameter  of  the  circle? 

188.  Two  secants  are  drawn  to  a  circle  from  an  outside 
point.  If  their  external  segments  are  10  and  6,  while  the  in- 
ternal segment  of  the  first  is  5,  what  is  the  internal  segment 
of  the  latter? 

189.  The  sides  of  a  A  are  x  y  =  4,  x  2  =  Q,  y  z  ==  S. 
Find  the  length  of  the  bisector  of  Z  -^. 

240. 

Proposition  XL,. 


Given:     A  B  C  D  any  inscribed  trapezium  with  A  C  and 
B  D  the  diagonals. 

To  compare  the  product  of  the  diagonals  with 
the  sum  of  the  products  of  the  opposite  sides. 


PROPORTION—SIMILAR  FIGURES.  163 

Sug.  1.     Draw  D  E  so  that  ZADE=ZBDC. 
Can  you  prove  (1)  As  BCD  and  A  B  D  similar  ? 

(2)  D  C  :  D  B  rr:  E  C  :  A  B  ? 

I.  ABDC^DBEC? 
How  much  of  the  required  result  has  been  found  ? 
Sug.  2.     Compare  As  A  D  E  and  B  D  C. 
Can  you  prove  II.  AD-BC— DBAE? 
What  results  if  I.  and  II.  are  added  and  factored  ? 
Generalize  the  truth  reached. 

241. 

Definition:  A  straight  line  is  said  to  be  divided  by  a 
given  point  in  extreme  and  mean  ratio  when  one  of  the  seg- 
ments is  a  mean  proportional  between  the  whole  line  and  the 
other  segment. 


The  straight  line  A  B  is  divided  internally  at  C  in  extreme 
and  mean  ratio  when  A  B  :  A  C  =  A  C  :  B  C.  It  is  divided 
exter7ially  at  D  in  mean  and  extreme  ratio  when  A  B :  A  D  = 
A  D  :  B  D. 

242. 

Proposition  XLI. 
Problem.      To  divide  a  line  in  extreyne  and  mean 
ratio.  ^ "  '  ~ "" "  - 


/ 

aE 

/                     -  "    \ 

1                   c   -              . 

/ 

ox- 

/ 

'"     \ 

/ 

,^-^'' 

Given:     Straight  line  A  B. 

To  Divide:     A  B  in  extreme  and  mean  ratio. 

Construction:     Let  B  C  be  a  perpendicular  to  A  B  at  B^ 


164  PLANE  GEOMETRY,  BOOK  III. 

and  made  equal  to  ^  A  B.  With  C  B  as  a  radius,  describe  the 
circle  C,  and  draw  the  line  A  C  cutting  the  circumference  at 
D  and  E.  Make  A  F  nr  A  D,  and  produce  B  A  making  A  G 
=  AE. 


/• 


i  A-  ; 

OX- 


Sug.     What  is  the  relation  between  A  B  and  A  E? 
Can  you  show — 

(1)  AE:AB  =  AB:AF? 

(2)  AE- AB:AB=-AB-  AF:AF? 

(3)  AF:AB  =  FB:AF? 

(4)  AB:AF  =  AF:FB? 
What  can  you  say  of  (4)  ? 

From  (1),  by  composition  (§  185),  show — 

(ly     AE  + AB:AE::AB-f  AF:AB. 

(2/     BG:AE  =  AE:AB. 

(3/     BG:AG=AG:AB,orAB:AG  =  AG:BG. 

What  can  you  say  of  (3)7 

(Study  this  proposition  carefully;  it  will  be  needed  in 
future  work.) 

243. 

Definition:    If  a  straight  line  is  divided  internally  and  ex- 
ternally in  the  same  ratio,  it  is  said  to  be  divided  harmonically, 

A  C  B  D 

r— ^ , T 

If  line  A  B   has  C  and   D  located  so  that  A  C  :  C  B  = 
A  D  :  B  D,  it  is  divided  harmonically. 


PROPORTION— SIMILAR  FIGURES. 


165 


244. 

Proposition  XLII. 


H 


^/ 


A-^ 


:xr 


Given:     The  straight  line  A  B. 

To  divide  it  harmonically  in  a  given  ratio,  as  in  the  ratio 
of  lines   X  and^. 

ConstrticHon.     Draw  A  H  making  any  convenient  angle 
with  A  B. 

Measure  off  A  D  =jv,  and  D  E  =  D  F  =  ^. 

Join  B  E,  B  F,  and  through  D  draw  D  G  ||  F  B  and  DC 
I  E  B  meeting  A  B  produced  in  C. 

Can  you  prove  {1)AG:GB  =  AD:DF  =y  :  x} 
(2)  A  C  :  B  C  =  A  D  :  E  D  =jK  :  ;»^? 

What  conclusion  can  you  draw?     Is  A  B  divided  har- 
monically in  ratio  of  jk  and  x} 

Exercise. 

190.     Can  you  prove  that  G  C  is  divided  harmonically  at 
the  points  A  and  B  ? 


166 


PLANE  GEOMETRY,  BOOK  III. 


245. 


Proposition  XLIII. 


Problem,  Upon  a  given  line^  homologous  to  a 
given  side  of  a  given  polygon^  to  construct  a  po  lygon 
similar  to  the  given  polygon. 


Given:     The  line  A'  B'  and  the  polygon  P. 

To  Construct — On  A'  B',  homologous  to  A  B,  a  polygon, 
F,  similar  to  P. 

Sug.  Draw  the  diagonals  A  C  and  A  D.  Under  what 
conditions  will  A  A'  3'  C  be  similar  to  A  A  B  C?     §205. 

When  will  P  and  P'  be  similar  ?     §205. 

Supplementary  Exercises. 


191.  To  inscribe    in  a  given  circle    a    A  similar  to   a 

given  A- 

192.  To  circumscribe  about  a  given  circle  a  A  similar 
to  a  given  A- 


AREAS.  167 


BOOK  IV. 

AREAS. 

Quadrilaterals. 

(1)  What  is  meant  by  a  unit  of  length?  a  unit  of  sur- 
face ?  a  unit  of  vohime  ?     Give  examples  of  each. 

(2)  What  is  meant  by  a  line  5  yards  long?  a  surface 
containing  25  square  feet?  a  volume  of  125  cubic  feet?  Give 
other  examples. 

(3)  Illustrate  the  difiference  between  equal  and  equiva- 
lent figures. 

(t)     What  is  meant  by  the  base  of  a  polygon  ?     Illustrate. 
(5)     What  is  meant  by  the  altitude  of  a  polygon  ?     Illus- 
trate. 

246. 

Proposition  I. 

Draw  2  parallelograms  on  equal  bases  and  having  equal 
altitudes.     Write  the  steps  to  show  how  they  are  related. 

247. 

Cor.  Using  the  same  figures,  can  you  compare  2  As 
having  equal  bases  and  equal  altitudes? 

Exercises. 

193.  What  is  the  path  of  the  vertex  of  a  A  of  constant 
area  on  a  fixed  base  ? 

191.  Draw  a  line  from  the  vertex  of  a  A  to  the  middle 
of  the  opposite  side.     How   does  it  divide  the  A  ?     Prove. 


168 


PLANE  GEOMETRY,  BOOK  IV. 


Draw  lines  from  vertex  to  points  which  are  distant  J,  J,  ^  of 
the  base  from  either  extremity  of  it.     Compare  these  As. 

195.  Draw  a  parallelogram,  A  B  C  D.  Join  B  D.  Take 
any  point,  P,  on  B  D  and  join  with  A  and  C.  Compare  /\s 
D  P  C  and  D  P  A. 

196.  Draw  2  As,  A  B  C  and  A'  B'  C,  in  which  B  C  =  B'  C 
and  AC  =  A'  C  and  Z  C  =  the  supplement  of  Z  C.  Can 
you  prove  the  As  equivalent  ? 

197.  ■  Construct  A  A  B  C,  then  on  the  same  base  con- 
struct a  rectangle  having  twice  the  area  of  the  A-     Prove. 


248. 

Proposition  II. 


Fig.  1. 


A  G  =  the  common  linear  unit  of  measurement.  . 

Gi^en:  The  two  rectangles  B  D  and  B  F  with 
equal  altitudes,  but  unequal  bases.  Compare  the 
rectangles. 

Case  /.  Let  the  base  A  D  contain  the  unit  AGS  times, 
and  the  base  A  F  contain  the  unit  3  times.  Write  an  equa- 
tion expressing  the  relation  of  A  F  to  A  D.  Call  it  (1).  At 
the  points  of  division  erect  Is  to  B  C.    Into  whatis  B  D  divided? 


AREAS.  169 

B  F  ?     Express  the  relation  between  these  two  rectangles  by 
an  equation.     Call  it  (2).     Compare  (1)  and  (2). 


r^ 


/ 


Given:  The  rectangle  m  ?i  and  m  n'  with  equal  altitudes? 
Suppose  />  ?z  =  1  dm.  and  />'  n  =  ^  dm.,  using  the  mm.  as  the 
unit,  what  is  the  ratio  of  these  bases?  Erect  Is  at  the  ex- 
tremities of  each  mm.  Compare  the  areas  ofmn  and  m'  nl* 
Are  the  bases  commensurable  ? 

In  Fig.  1,  could  the  relation  between  B  D  and  B  F  be  ex- 
pressed if  we  used  ^  A  G  as  the  common  unit?  Does  the  re- 
lation between  B  D  and  B  F  change  if  the  unit  is  halved? 
What  effect  does  it  have  on  the  small  rectangles  when  we  de- 
crease the  size  of  the  common  unit?  Could  we  express  the 
relation  between  B  D  and  B  F  if  we  took  J  A  G  as  a  common 
unit?  How  small  a  fractional  part  may  we  take  and  still  ex- 
press the  true  relation  ? 

Suppose  we  think  of  the  common  unit  as  1  millionth  of 
A  G,  how  many  rectangles  may  be  formed  in  B  F  ?  in  B  D  ? 
in  ED? 

{Review  on  Principles  of  Limits. — {Review  §  195.) 


B 

^ — 


o      tt 


I^et  A  M  and  A'  M'  be  two  equal  variables  which  con- 
stantly =^  *  A  B  and  A'  B'  respectively.  I^et  us  compare  A  B 
and  A'  B'.  If  possible,  suppose  A  B  >  A'  B',  measure  off  on 
A  B  a  distance  A  C  =  A'  B'.    What  is  the  limit  A  M  approaches  ? 

(*  =  means  approach  as  a  limit.) 


170 


PLANE  GEOMETRY,  BOOK  IV. 


May  the  limit  of  A  M  pass  A  C?  What  is  the  limit  A'  M'  ap- 
proaches? Can  A'  M'  ever  reach  A'  B'  or  A  C?  What  is  the 
relation  of  A  M  and  A'  M?  Do  you  see  any  absurdity  in  sup- 
posing AB  >  A'B?  Now  suppose  that  AB  <  A'B'  and  let 
A'  B'  be  measured  off  on  A  B  produced  and  let  A  D  =  A'  B'. 
What  will  A  M  =  ?  A'  M'  =?  Can  A'  M'  become  greater  than 
A  B?  Can  A  M  become  equal  to  A  B  ?  How  are  the  variables 
A  M  and  A'  M'  related?  What  absurdity  by  our  last  supposi- 
tion ?  Now  if  A  B  cannot  be  greater  than  A'  B'  and  it  cannot 
be  less  than  A'  B^  what  condition  must  exist?  Write  a  general 
statement  of  what  we  have  proved  and  memorize  it.  J 

Case  II. 


u 

,                            A. 

9 

u                          » 

Let  D  C  and  D  F  in  the  2  as  A  C  and  A  F  be  incommen- 
surable, but  Jiaving  same  altitude.  Suppose  a  unit  of  length, 
u,  is  contained  an  exact  number  of  times  in  D  C,  say  3  times,  and 
in  D  G  once,  with  remainder.  What  do  we  say  of  D  C  and 
D  G?  Erect  a  1  at  G  forming  the  □  A  G.  How  are  A  G  and 
A  C  related?  Now  imagine  the  unit  of  comparison  to  de- 
crease.    How  does  it  affect  D  G?    G  F  ?     What  does  D  G  =? 


What  does  AG=^?   Now  do  you   see   that 


AG     DG 


Do 


AC     D  C 

you  see  that  each  member  of  the  equation  is  a  variable? 
Does  each  approach  a  limit  ?  Are  they  always  equal?  What 
have  we  learned  about  two  equal  variables  as  they  approach 
limits?     Draw  conclusion.     Call  it  Prop.  II. 

249. 

Which  side  of  a  rectangle  may  be  considered  the  base? 
Can  A  D  be  considered  the  base  of  each  of  the  rectangles, 


AREAS. 


171 


A  F   and   AC?     Can  you  then  state  a   corollary   from  the 
preceding  proposition  ? 


250. 


Proposition  III. 


I 

~  "1 

a  i 

c 

1- 

- 1 

b' 


Given  the  rectangles  A  and  B  with  altitudes  a  and  a  and 
bases  d  and  d'.  Let  C  be  a  third  rectangle  with  altitude  a  and 
base  d'.  Compare  A  and  C  in  the  fractional  form.  Call  the 
equation  (I).  Compare  Cand  B  in  the  same  manner  as  in  (1), 
call  new  equation  (2).     Multiply  (1)  by  (2). 

What  does  the  new  equation  show? 

Write  a  general  statement  and  call  it  Prop.  III. 

Exercises. 

198.  Find  the  ratio  of  a  rectangular  field  72  rods  by  49 
rods  to  a  board  18  inches  by  14  inches. 

199.  A  rectangular  court-yard  is  18>^  yards  by  1^}^ 
yards.  Find  the  relation  of  its  surface  to  a  rectangular  paving- 
stone  31  inches  by  18  inches. 

200.  On  a  certain  map  the  scale  is  1  inch  to  10  miles. 
How  many  acres  are  represented  on  this  map  by  a  square 
whose  perimeter  is  1  inch? 

201.  A  square  and  rectangle  have  the  same  perimeter, 
100  yards;  the  length  of  the  rectangle  is  4  times  its  breadth. 
Compare  the  areas. 


172  PLANE  GEOMETRY,  BOOK  IV. 

251. 
Proposition  IV. 

What  is  meant  by  finding  the  surface  of  a  plane  figure  ? 
the  numerical  measure  ? 


s 


Given:  The  rectangle  A  and  the  square  B.  I^et  B  be 
the  unit  of  surface.  How  are  these  two  figures  related?  Ex- 
press this  relation  in  a  fractional  form;  clear  of  fractions. 
What  is  B? 

What  then  does  the  area  of  A  equal?  What 
does  this  prove  about  the  area  of  any  rectangle? 

Write  the  general  truth  and  call  it  Prop.  IV. 

252. 

As  a  corollary  to  this  proposition,  show  how  to  find  the  area 
of  a  square.  Suppose  the  sides  of  the  rectangle  are  multiples  of 
the  linear  unit,  can  you  show  any  easy  way  to  illustrate  the 
area  of  thejrectangle? 

[Remark:  Fectangle  and  triangle  are  often  used  for 
their  areas.  The  prodiict  of  two  lines  means  the  product  of 
their  numerical  measure.  The  unit  of  surface  means  a  square, 
each  side  of  which  is  a  U7iit  of  length.  The  measurement  of 
a  surface  is  the  number  of  times  it  contains  the  unit  of  surface. 
It  equals  the  product  of  the  numerical  measures  of  the  base  and 
altitude.! 


AREAS. 


173 


253. 

Proposition   V. 


Fig.  1.  Fig.  2. 

Given  rectangles  AC,  AD  in  Fig.  1,  and  A  D,  A  C  in 
Fig  2.  In  Fig.  1  write  an  expression  for  B  D.  The  sum  of 
what  rectangles  =  A  D?  The  product  of  what  lines  =  AC? 
ED?  The  product  of  what  lines  =  A  D  ?  Place  these  three 
products  into  an  equation.  Can  you  interpret  this  equation 
in  a  general  way?  What  does  B  D  =  in  Fjg.  2?  What  does 
it  =  in  Fig.  1  ?  What  does  A  B  •  B  D  =r  in  Fig.  1?  in  Fig.  2  ? 
Write  the  equivalent  products  for  A  B  •  B  D  in  Fig.  1,  and  just 
under  this  equation  write  the  equivalent  products  of  A  B  •  B  D 
in  Fig.  2.  If  we  let  a,  b,  c  be  the  numerical  measure  of  A  B, 
B  C,  C  D  respectively,  then  by  substitution  in  the  last  two 
equations  we  get  a  {b  ±  c)  ^=^  a  b  ±  a  c. 

254. 

Cor.  I.  Let  M  and  N  each  represent  a  line.  (M  +  N)* 
equivalent  to  rectangle  (M  -f-  N)  (M  -h  N)  equivalent  to  rect- 
angle M  (M  +  N)  -f  N  (M  +  N)  equivalent  to  M^  +  rectangle 
M  N  -|-  rectangle  M  N  +  N^  equivalent  toM~  -|-  N'  +  2  rect- 
angle M  N.  Let  ?n  and  7i  be  the  numeral  measure  of  M  and  N. 
What  algebraic  expression  represents  the  areas  ?  Let  M  —  N 
be  the  difference  of  two  lines;  work  out  the  value  of  (M  — N)". 
Write  a  general  statement  for  (M  ±  N)^  in  which  M  and  N 
represent  lines. 

255. 

Cor.  II.  In  same  general  way  write  out  the  value  of  (M 
+  N)  (M  — N). 

Construct  figures  to  illustrate  each  of  the  above  cor- 
ollaries. 

May  some  algebraic  expressions  have  a  geometrical  inter- 
pretation ?     Illustrate. 


174 


PLANE  GEOMETRY,  BOOK  IV. 


Exercises. 

202.  Upon  thediagoaal  of  a  rectangle  24  feet  X  10  feet 
a  A  equivalent  to  the  rectangle  is  constructed.  Find  its 
altitude. 

203.  The  area  of  a  rectangle  is  3456  square  inches  and 
the  base  is  2  yards.    Find  perimeter  in  feet. 

204.  A  rectangle  18  X  6.  Find  the  side  of  an  equiva- 
lent square. 

256. 

Proposition  VI. 

Compare  the  rectangle  and  rhomboid  having  the 
same  base,  of  any  length,  and  equal  altitudes  in  each 
of  the  following  cases  : 


Fig.  1. 


Fig.  3. 

(1).     Suppose  X  to  fall  between  F  and  B. 

(2).     Suppose  X  to  fall  on  B. 

(3).     Suppose  X  to  fall  to  the  right  of  B. 


AREAS.  175 


Fig.  4. 

In  Fig.  4,  F7  and  A  D'  are  parallel,  and  A  B  is  any  rect- 
angle, and  A  D  =:  A'  D'. 

Compare  the  rhomboid  A.' y  with  the  rectangle 
AB. 

Write  a  statement  of  what  has  been  proved  in  the  four 
cases  above.     Call  it  Prop.  VI. 

257. 

Cor.  I.  Show  that  two  parallelograms  having 
the  same  base  and  equal  altitudes  are  equivalent. 

258. 

Co^.  II,  Show  that  two  parallelograms  having 
equal  bases  and  equal  altitudes  are  equivalent. 

259. 

Cor.  Ill  Show  that  two  parallelograms  having 
equal  altitudes  are  to  each  other  as  their  bases ;  two 
parallelograms  having  equal  bases  are  to  each  other 
as  their  altitudes ;  and  any  two  parallelograms  are 
to  each  other  as  the  products  of  their  bases  by  their 
altitudes. 

260. 

Cor.  IV.  Show  how  to  find  the  area  of  any 
parallelogram. 


176  PLANE  GEOMETRY,  BOOK  IV. 

261. 

Cor.  V.  Can  you  show  how  to  find  the  area  of 
any  triangle? 

262. 

Cor.  VI,  Can  you  show  that  triangles  with 
equal  bases  and  equal  altitudes  are  equivalent  ? 

263. 

Cor.  VII.  Can  you  show  that  triangles  with 
equal  altitudes  are  to  each  other  as  their  bases ;  and 
those  with  equal  bases  are  to  each  other  as  their 
altitudes? 

264. 

Cor,  VIII.  Can  you  show  that  any  two  trian- 
gles are  to  each  other  as  the  products  of  their  bases 
and  altitudes. 

Exercises. 

205.  The  altitude  and  base  of  a  A  being  35  and  12,  re- 
spectively, what  is  its  area  ? 

206.  The  area  of  a  A  is  221  square  feet ;  its  base  is  52^ 
yards.     What  is  its  altitude  in  inches? 

207.  The  bases  of  two  parallelograms  are  15  cm.  and  16 
cm.  respectively;  and  their  altitudes  are  8  cm.  and  10  cm. 
respectively.     What  is  the  ratio  of  their  areas? 

208.  Two  As  of  equal  areas  have  their  bases  26  mm.  and 
36  mm.  respectively.    What  is  the  ratio  of  their  altitudes? 

209.  Draw  any  straight  line  through  the  point  of  inter- 
section of  the  diagonals  of  a  parallelogram  terminating  in  a 
pair  of  opposite  sides  and  show  how  the  parallelogram  is 
divided. 


AREAS.  177 

210.  If  E  is  the  middle  point  of  C  D,  one  of  the  non- 
parallel  sides  of  the  trapezoid  A  B  C  D,  prove  that  the  triangle 
ABE  [draw  A  E  and  B  E]  is  equivalent  to  one-half  the 
trapezoid. 

211.  If  E  and  F  are  the  middle  points  of  the  sides  A  B 
and  A  C  of  a  triangle,  and  D  is  any  point  in  B  C,  show  how 
the  quadrilateral  A  E  D  F  is  related  to  the  A  A  B  C. 

212.  Join  the  middle  points  of  the  adjacent  sides  of  any 
quadrilateral.  What  is  the  new  figure  ?  How  is  it  related 
to  the  quadrilateral  ? 

213.  If  two  equivalent  As  have  a  common  base  and  have 
their  vertices  on  opposite  sides  of  the  base,  the  line  joining 
their  vertices  is  bisected  by  the  base  (produced  if  necessary). 

214.  Construct  a  parallelogram,  A  B  C  D,  and  draw  the 
diagonal  A  C.  Take  any  point,  P,  on  A  C  and  join  it  with 
B  and  D.  Compare  the  areas  of  A  B  P  and  A  D  P,  of  B  P  C 
and  D  P  C. 

265. 

Proposition  VII. 


What  is  the  figure  A  B  C  D  if  A  B  is  ||  D  C  ?  What  is 
the  altitude?      W/iai  is  its  area  f 

[//zw/.— What  2As  compose  A  BCD?  ABD  =  ?  BCD 
==?     Therefore  A  B  C  D  =  ?] 

Generalize  this  equation  and  call  it  Prop.  VII. 


13 — 


178  PLANI3  GEOMETRY,  BOOK  IV. 

266. 

Cor.  J.  Show  that  one-half  the  sum  of  the  parallel  bases 
equals  the  median  of  the  trapezoid? 

ExKRCiSES. 

215.  Altitude  of  ^  trapezoid  is  5,  bases  8  and  10,  find  area^ 

216.  Construct  an  irregular  pentagon,  and,  having  com- 
passes and  rule,  show  how  to  compute  area. 

217.  The  area  of  a  rhombus  =  one-half  the  product  of 
its  diagonals.     Prove. 

218.  Rough  boards  are  usually  narrower  at  one  end  than 
at  the  other,  for  which  reason  the  lumber  merchant  usually 
measures  their  width  in  the  middle.  Can  you  explain  the 
principle  involved  in  such  measurement  ? 

219.  A  carpenter  wishes  a  trapezoidal  board  whose  non- 
parallel  sides  must  be  equal.  He  lays  off"  equal  angles  with 
one  of  the  bases  and  saws  out  his  board. 

[Let  the  sides  of  the  Z  s  which  coincide  with  the  base  of 
the  trapezoid  extend  in  opposite  directions  from  the  vertices.] 
Prove  that  his  method  is  right. 

220.  Suppose  the  trapezoidal  board  mentioned  in  Kx. 
219  simply  required  that  the  base  angles  made  by  the  non- 
parallel  sides  should  be  equal.  What  could  the  carpenter  dis- 
cover concerning  the  non-parallel  sides  ? 

221.  (1)  Can  you  give  two  methods  of  finding  the  area 
of  any  polygon  ? 

(2)     Show  that  equiangular  As  are  similar. 


AREAS. 

267. 

Proposition  VIIL 
Areas  of  Polygons — Continued. 


17» 


Fig. 


Given:    The  similar  As  A  B  C  and  A'  B'  C. 

To  Show — That  corresponding  altitudes  are  to 
each  other  as  any  two  homologous  sides. 

Sug.  If  C  D  and  C  D'  are  homologous  altitudes,  can  you 
show  that  A  C  D  is  similar  to  A'  C  D'? 

[When  are  triangles  similar?] 

CanyoushowthatCD:C'D'  =  AC:A'C'  =  AB:  A'B'  = 
B  C  :  B'  C? 

Write  the  general  truth  as  Prop.  VIII. 

Scholium:  The  ratio  of  any  two  homologous  sides  of 
similar  polygons  is  called  the  ratio  of  similitude. 

222.  Can  you  show  that  any  two  similar  As  are  to  each 
other  as  the  squares  of  any  two  homologous  lines,  or  are  in 
the  ratio  of  similitude  of  the  triangles  ? 

223.  In  Fig.,  §  267,  if  A  B  =  10,  A'  B'  -=  6  and  area  A 
A'  B'  C  =  36,  what  is  the  area  of  A  B  C? 


180  PLANE  GEOMETRY,  BOOK  IV. 

268. 

Proposition  IX. 


Given:  As  A  B  C  and  A  B'  C  any  2  As  having  an  angle 
of  one  equal  to  an  angle  of  the  other. 

To  Compare— k  B  C  and  A  B'  C^ 

Sug,  Compare  ABC  and  A  B'  C.  Compare  A  B'  C  and 
A  B'  C.  Express  these  two  comparisons  in  fractional  forms- 
Multiply  the  two  equations  together  and  simplify  the  result- 
Express  the  result  in  a  general  statement.     This  is  Prop.  IX. 

269. 

Cor.  /.  If  2  parallelograms  have  an  angle  of 
the  one  equal  to  an  angle  in  the  other,  how  are  they 
related  ? 

224.  Given  the  perimeter  of  a  triangle  and  the  radius 
of  the  inscribed  circle  to  find  its  area. 


270. 


Cor,  II,  If  the  products  of  the  sides  about  the 
equal  angles  are  equal,  what  can  you  say  of  the 
triangles? 


AREAS.  181 

271. 
Proposition  X. 


Fig. 

Given:  A — C  and  A' — C,  similar  polygons,  and  let  S  and  S' 
represent  the  areas  respectively.  Draw  the  diagonals  B  B, 
E  C  and  E'  B'  and  E'  C.  (See  Ex.  under  Prop.  VIII.)  Can 
you  form  an  equation  between  i\s  ABE  and  A'  B'  E'  on  the 
sides  A  Band  A'  B'?  Similarly  compare  E  B  C,  E'  B'  C  and 
E  C  D  and  E'  C  D'.  What  can  you  say  of  the  homolo- 
gous sides  of  the  polygons  ?  A  B  :  A'  B'  =  B  C  :  B'  C 
=  C  D  :  C  D',  etc.  But  in  a  proportion  like  powers  nre 
in    proportion,  §  182;   hence  A~b'  :  A'  #    =  ¥c'  :  B^'= 

8        — a 

CD  :  C  D  .  Now  substitute  these  values  in  your  first 
equations.  By  proportion,  §188,  the  sum  of  all  the  ante- 
cedents is  to  the  sum  of  all  the  consequents  as  any  ante- 
cedent is  to  its  consequent.  Can  you  write  an  equation 
so  that  the  sum  of  the  As  in  the  first  figure  shall  be  to  the 
sum  of  the  As  in  the  second  as  any  A  in  first  is  to  any  corre- 
sponding A  io  second?  Now  substitute  for  the  ratio  of  the  As 
its  value  above.  Substitute  for  the  sums  of  the  As  in  A  —  C 
and  A'  —  C,  S  and  S'  in  your  last  equation.  Generalize  and 
call  the  statement  Prop.  X. 

272. 

Cor.  I.  Can  you  show  that  similar  polygons  are 
to  each  other  as  the  squares  of  any  homologous  lines? 


:  J  PLANE  GEOMETRY,  BOOK  IV. 

273. 

Cor,  //.  Prove  that  the  homologous  sides  of 
any  two  similar  polygons  are  as  the  square  roots  of 
the  areas  of  those  polygons. 

274. 

Proposition  XI. 
In  the  rt.  L\  A  B  C,  let  A  B  be  the  hypotenuse.  Erect 
square  A  B  E  F  on  A  B.  Upon  the  legs  A  C  and  B  C  con- 
struct squares  ACGH,BCKL,  respectively,  and  join 
H  B,  F  C  and  draw  C  D  1  to  A  B,  and  produce  it  to  cut  E  F 
at  m.  Can  you  prove  that  A  B  A  H  is  equal  to  A  C  A  F? 
Can  you  show  how  A  A  B  H  is  related  to  the  square  A  G? 
How  is  A  A  C  F  related  to  the  rectangle  AM?  Compare  the 
square  and  the  rectangle.  By  what  axiom  do  you  make  the 
comparison?  Join  A  and  L,  C  and  E.  Can  you  show  that 
the  square  C  L  is  equivalent  to  the  rectangle  D  E?  Add 
squares  C  I^  and  A  G.  How  does  the  result  compare  with 
that  obtained  in  §  225?  Read  the  short  biography  of  Pythag- 
oras in  the  notes. 

275. 
Cor,  /.     If  the  legs  of  a  rt.  A  be  given,  write 
an  equation   which    will  indicate  how  to  find  the 
hypotenuse. 

276. 

Cor,  II,     Draw  any  square,  A  B  C  D.     Can  you 

t  8  AC 

show  that  AC  =  2  A  B  ?     Also,  that  j-^  =  V2} 

What  does  this  equation  mean?     [§  §  172,  173.] 
Exercises. 

225.  Length  of  rectangle  60,  altitude  5.     Find  diagonal, 

226.  In  the  Fig.  of  Prop.  XI.  [§  274],  show  how  C  F 
and  B  H  are  related. 


EXERCISES.  183 

227.  Prove  that  lines  B  K  and  A  G  are  ||.     [§  274] 

228.  Show  that  the  sum  of  the  Is  from  H  and  t,  to  AB 
produced  is  equal  to  A  B.     [§  274] 

229.  Compare  A  K  C  G,  Z\  B  E  L,  and  A  C  G  K  with 
AA  BC.     L§274] 

230.  Prove  that  C,  H,  and  L  are  in  the  same  straight 
line.     [§274] 

231.  Construct  a  square  equivalent  to  the  sum  of  any 
number  of  squares. 

232.  Construct  a  line  whose  length  is  ^^"2^  "^Y,  V^"5^  ^~b, 
Vl    (in  two  ways),  V  li  (in  three  ways). 

233.  If  similar  polygons  are  constructed  on  sides  of  a 
right  A>  show  that  the  polygon  on  the  hypotenuse  is  equiva- 
lent to  the  sum  ot  those  on  the  other  two  side-. 

234.  In  Fig.,  §  271,  if  A  B  :  A'  B'  =  ;»; :  j,  what  will  be 
the  ratio  of  the  square  described  on  E  B  to  that  described  on 
E'B'? 

235.  In  the  same  figure,  \i  s  \  s  =  x  :  y,  what  is  the 
ratio  of  any  two  homologous  lines,  as  E  C  and  E'  C  ? 

236.  Find  the  area  of  a  right  isosceles  A.  if  the  hypot- 
enuse is  50  rods  in  length. 

237.  Two  parallel  chords  are  each  10  feet  in  length  and 
the  distance  between  them  is  8  feet.  What  is  the  radius  of 
the  circle  ? 

238.  A  rectangular  table  contains  26.4  square  feet;  its 
width  is  2.2  feet.     Find  the  length  of  its  diagonal. 

239.  If  one  angle  of  a  right  triangle  is  30°,  how  does  the 
hypotenuse  compare  with  the  side  opposite  the  angle  of  30°  ? 

240.  Construct  an  equilateral  A  on  each  side  of  a  right 
triangle.  Show  that  the  equilateral  A  on  the  hypotenuse  is 
equivalent  to  the  sum  of  those  on  the  other  two  sides. 

241.  One  angle  of  a  right  triangle  is  60°.  Construct  an 
equilateral  triangle  on  the  hypotenuse  and  compare  its  area 
with  the  rectangle  whose  sides  are  the  two  legs  of  the  right  A- 

242.  If  two  triangles  have  two  sides  of  one  equal  respect- 


184  PLANE  GEOMETRY,  BOOK  IV. 

ively  to  two  sides  of  the  other  and  the  included  angles  sup- 
plementary, how  are  the  areas  related  ? 

243.  The  side  of  an  equilateral  triangle  is  20  decimeters. 
What  is  its  altitude  ? 

244.  Prove  how  a  trapezoid  is  divided  by  a  line  joining 
the  mid-points  of  the  parallel  sides 

245.  (I)  On  a  given  line,  mn,  the  hypotenuse,  construct 
a  rt.  A  equivalent  to  a  given  A,-     When  is  this  impossible? 

(2)     Prove  that  3  times  the  square  of  the  side  of  an  equi- 
lateral A  equals  4  times  the  square  of  the  altitude. 

246.  If  the  acute  Z   B  of  the  rt.  A  A  B  C  is  double  the 


Z  A,  prove  that  AC    =  3  B  C   . 

247.     If  the  Z  A  of  the  A  ABC  above  is  30°,  prove  that 
area  ofABC^JABX  AC 

277. 
Proposition  XII. 


Given:  Any  quadrilateral,  A  B  C  D,  and  the  diagonals 
A  C,  B  D.  Call  the  centers  of  these  diagonals  E  and  F.  Join 
them. 

Can  you  show  that  the  sum  of  the  squares  of  the 
4  sides  is  equivalent  to  the  sum  of  the  squares  of 
the  diagonals  plus  4  times  the  square  of  the  line 
joining  the  middle  points  E  and  F? 


AREAS.  185 

[Hint. — Join  B  E  and  D  E.     By  §233  write  an  equation  for 

2 2 

A  B  -f  B  C  ,   call  it  equation  (1);  also  write  an  equation  for 
Ad'+^^'.     Call  this  (2).     Add  (1)  and  (2).     In  A  B  E  D 


what  does  B  E    +  D  E   equal?  2  B  E  +  2  D  E   =  ?     Can  you 
substitute  this  in  equation  (3)?     Show  that  4  AE^  =  2  ( AE) 
=A  C  .     Similarly  find  the  value  of  4  B  F    and  substitute.] 

Write  the  complete  statement  and  call  it  Prop.  XII. 

278. 

What  corollary  can  you  state  concerning  any  parallelo- 
gram? 

Let  A  —  area  of  an  equilateral  A  and  a  one  side.  Find 
area  in  terms  of  its  side. 

279. 

Proposition  XIII. 


Fig.  2. 
Give?i :     The  sides  of  any  triangle,  as  a,  b,  c. 

To  Find— I.     An  expression  for  the  altitude  in 
terms  of  the  sides. 

II.     The  area  in  terms  of  the  sides. 

Sug.     Let  k  be  the  altitude. 

By  §  231  can  you  show  that  <:'  =  a'  4-  ^'  —  2  a     C  D? 

What  is  the  value  of  C  D  ? 


186  PLANE  GEOMETRY,  BOOK  IV. 

By  substituting  and  factoring, 
>^':=^'— C3'.     Why? 

^  <i  a  ' 

(1)  4^^  ^'  —  {g}  ^  b^  —  cy    _ 

(2)  (2ad  +  a'  -^  d'  —  c')  (2  a  d  ~  a'  —  d'  +-  c')  _ 

4.  a'  "~ 

(3)  [(a  +  by-c^]    [c^-'(^—J)']_ 

(4)  {a-j-d^  c)  {a  ^b  —  c)  jc^a  —  b)  {c  ~  a  +  b>^  __ 

4a^  — 

[To  shorten  the  work,  l^t  a  -\-  b  -\-  c  =  2  S; 
Then  a^  b  —  c  =  2  S—  2  r  =  2  (S  -  c), 
^+^-.^  =  28  —  2^=2  (S  -b), 
c  —  a-^b  =  2S  —  2a  =  2{S  —  a). 
Now  substitute  these  values  in  (4),  and]  : 

(5)  2S  -2(8—^)  -2(8  —  /;)  -2(8  —  ^)^ 

.-.     h^-sl  S{S-a){S-b){S-c). 

Write  expression  II. 
Call  the  Prop.  XIII. 
Note. — The  solution  given  is  algebraic. 
248.     If  the  sides  of  a  triangle  are  10  dm.,  17  dm.,  21  dm., 
what  is  the  area  ?    What  is  each  altitude? 


AREAS.  187 

280. 
Proposition  XIV. 


Giveiv.  Any  triangle,  ABC,  with  sides  a,  b,  c,  and  alti- 
tude h.  Circumscribe  a  circle  about  the  triangle  and  draw  a 
diameter,  A  E,  through  the  vertex  A.  Join  E  with  the  other 
extremity  of  the  shortest  side. 

To  Find — The  area  of  the  triangle  in  terms  of  the  radius- 
Call  the  area  A. 

Sug.  b  c  ='i  [§  2B9]  Instead  of  A  E,  we  may  use 
2  r,  r  being  the  radius  of  the  circumscribed  circle,     h  a^=l 

:.ab  c  =  '>    A  =  ? 

Call  this  Prop.  XIV.     Write  it  carefully. 

249.  Find  the  diameter  of  the  circumscribed  circle  in 
Ex.  248. 

250.  By  using  the  value  of  h  in  §  279  and  using  §  280,  find 
the  value  of  the  radius  of  the  circumscribing  circle  about  a 
triangle  in  terms  of  the  sides  when  they  are  given. 


188 


PLANE  GEOMETRY,  BOOK  IV. 

281. 
Proposition  XV. 


Given:     Any  triangle,  ABC,  with  the  median,  m^  from 
vertex  A  and  the  sides  a,  b,  c. 

To  Find — An  expression  for  m  in  terms  of  the  sides. 

Sug.     By  §  233  ^^  +  r^  -  2  w'  -f  2  (|)'. 

Find  m.     Draw  the  median  from  the  vertices  C  and  B  and 
conipare  the  three  expressions  for  the  medians. 


282. 
Proposition  XVI. 


Fig. 

Given:     Any   triangle,  A  B  C,  with  the  sides  a,  b,  c  and 
the  bisector  of  the  angle  at  C,  C  D  ==  r. 

To  Find— Am  expression  for  the  bisector,  g,  in   terms  of 
the  sides. 


AREAS.  189 

Szig.     Circumscribe  a  circle  about  the  triangle. 
From  §  288  (I)  a  ^  =     A  D  •   D  B  +  g'; 
.-.    (2)  ^^  ==     «  ^  —  A  D  .   D  B. 
(3)  A  D  :  D  B  =  ^:a.     Why? 
(4)AD+BD:AD  =  /^-h^:^.     Why  ? 
(5)  A  D  +  B  D  :  ^  4-  «  =  A  D  :  /^.     Why  ? 

==  D  B  :  «.     Why? 

(7)  What  is  the  value  of  A  D  ?     Of  D  B? 

Call  a  -\-  b  ^  c,2s,  then  a  -\-  b  —  c  =   2s  — 2c. 

/ox   oi  1  2         ab  —  2s-2(s—e) 

(8)  Show  that  g^  =   ^^-p^J, 

2 


Write  the  proposition. 

251.  Find  an  expression  for  the  bisector  of  Z  B  in  §  282. 

Supplementary  Exercises. 

252.  If  the  sides  of  an  isosceles  triangle  are  denoted  by 

a,  a,  and  b  respectively,  prove  that  its  area  =  -.  >!  Aa^ b*. 

4 

253.  The  area  of  an  isosceles  right  triangle  is  equal  to 
one-fourth  the  area  of  the  square  described  upon  the  base. 

254.  Three  times  the  square  of  the  side  of  an  equilateral 
triangle  is  equal  to  four  times  the  square  of  the  altitude. 

255.  If  the  acute  angle  B  of  the  right  triangle  A  B  C  is 

8  2 

double  the  angle  A,  prove  that  A  C    =  3  B  C  . 

256.  If  the  angle  A  of  the  triangle  A  B  C  is  30°,  prove 
that  area  ABC  =  JABXAC. 

257.  If  E  is  any  point  in  the  side  B  C  of  the  parallelo- 
gram A  B  C  D,  the  triangle  A  E  D  is  equivalent  to  one-half 
the  parallelogram. 

258.  If  E  is  any  point  within  the  parallelogram  A  B  C  D, 
the  triangles  ABE  and  C  D  E  are  together  equivalent  to  one- 
half  the  parallelogram. 


190  PLANE  GEOMETRY,  BOOK  IV. 

259.     If  A  D  is  the  perpendicular  from  A  to  the  side  B  C 


of  the  triangle  ABC,  prove  that  AB  —  AC   =BD  — CD. 

260.     If  D  is  the  intersection  of  the  perpendiculars  from 
the  vertices  of  the  triangle  A  B  C  to  the  opposite  sides,  prove 


that  AB— AC=BD— CD. 

261.  The  area  of  a  rhombus  is  24  and  its  side  is  5.  Find 
the  lengths  of  its  diagonals. 

262.  If  one  diagonal  of  a  quadrilateral  bisects  the  other, 
it  divides  the  quadrilateral  into  two  equivalent  triangles. 

263.  Any  straight  line  drawn  through  the  point  of  inter- 
section of  the  diagonals  of  a  parallelogram,  terminating  in  a 
pair  of  opposite  sides,  divides  the  parallelogram  into  two  equiv- 
alent quadrilaterals. 

264.  The  sum  of  the  squares  of  the  lines  joining  any 
point  in  the  circumference  of  a  circle  with  the  vertices  of  an 
inscribed  squa  e  is  equal  to  twice  the  square  of  the  diameter 
of  the  circle      (§204.) 

265.  If  D  is  the  middle  point  of  the  side  B  C  of  the  trian- 


gle A  B  C  right-angled  at  C,  prove  that  A  B  —A  D   =  3  C  D  . 

266.     If  D  is  the  middle  point  of  the  hypotenuse  A  B  of 
the  right  triangle  ABC,  prove  that 


CD   —KAB    +BC    +CA). 
267.     If  a  line  is  drawn  from  the  vertex  C  of  an  isosceles 
triangle  meeting  the  base  A  B  produced  at  D,  prove  that 


C  D  ~C  B        AD  X  BD. 

268.  If  A  D  is  the  perpendicular  from  one  of  the  extrem- 
ities  of  the  base  A  C  to  the  opposite  side  in  the  isosceles  trian- 
gle ABC,  prove  that 


3AD   +2BD    +CD    =AB   -[-BC    -hCA. 

269.  One  of  the  equal  sides  of  an  isosceles  triangle  is  18 
dkm.  and  the  base  is  30  m.     Find  the  area  of  the  triangle. 

270.  One  of  the  sides  of  an  equilateral  triangle  is  20  m, 
Find  its  area. 


EXERCISES.  191 

271.  The  sides  ofa  A  are  20  dm,  30  dm.,  40  dm.  Find  the 
length  of  the  bisectors  of  the  angle. 

2/2.  If  the  sides  of  a  triangle  are  9  m.,  U  m.,  12  m., 
what  is  the  diameter  of  the  circumscribed  circle? 

273.  The  two  sides  of  a  parallelogram  are  a,  and  b,  and 
one  diagonal  is  e.     What  is  the  length  of  the  other  diagonal? 

274.  The  hypotenuse  of  a  right  isosceles  triangle  is  a. 
Find  its  area. 

275.  The  diagonals  of  a  rhombus  are  to  each  other  as 
4  :  7  and  their  sum  is  16.      Find  the  area. 

276.  What  is  the  area  of  a  triangle  two  of  whose  sides 
are  &  dm.  and  12  dm.,  and  the  angle  included  between  them 
is  30°? 

277.  Two  sidesof  a  triangle  are  a  and  b,  and  the  included 
angle  is  30°.    What  is  the  area? 

278.  In  the  last  example,  suppose  the  included  angle, 
were  150°,  what  would  be  the  area? 

279.  If  the  two  sides  of  a  triangle  are  a  and  b,  and  the 
included  angle  is  45°  or  135°,  can  you  show  that  the  area  is 

\ab  ro. 

280.  In  the  last  example,  if  the  included  angle  is  60° 
or  120°,  the  area  \s\a  b\I'6.     Prove. 

281.  The  area  of  a  trapezoid  is  46  ares,  and  its  bases 
are  97  m.  and  133  m.     Can  you  find  its  altitude? 

Problems  in  Construction. 

282.  Construct  a  :3,  A  B  C  D,  with  D  K  the  altitude  and 
A  B  the  base  of  the  a.  Find  a  mean  proportional  between 
the  base  and  the  altitude  of  A  B  C  D.  Construct  a  square 
equivalent  to  the  a. 

283.  CoUvStruct  a  square  equivalent  to  a  given  A* 

284.  Find  a  fourth  proportional  to  these  lines  V!}l 


285.     Given  rectangle  A  B  C  I)  and  the  line  E  F.     Con- 
3truct  a  rectangle  on  E  F  having  the  area  of  rectangle  A  B  C  D, 


192 


PLANE  GEOMETRY,  BOOK  IV. 

283. 

Proposition  XVII. 


Problem.      To  construct  a  triang/e  equivalent  to  a 
given  polygon. 


Let  A  B  C  D  E  be  any  polygon.  Let  B  F  be  ||  diagonal 
A  C,  and  E  G  ||  diagonal  A  D.  Compare  A  A  B  C  and  A 
AFC,  also  A  A  E  D  and  A  A  G  D.  Compare  /^  A  F  G  with 
polygon  A  B  C  D  E. 

Exercises. 

286.  Can  you  construct  a  square  equivalent  to  a  given 
polygon? 

287.  Construct  a  rectangle  upon  a  given  line,  A  B,  equiv- 
alent to  a  given  polygon. 

284. 

Proposition  XVIII. 


Given  the  square  m  and  the  line  A  B. 

Can  you  construct  a  rectangle  equivalent  to  m^ 
having  the  sum  of  its  base  and  altitude  equal  to  A  Bf 


PROBLEMS  OF  CONSTRUCTION.  193 

YHint. — (1)  Assume  the  problem  solved  and  then  study 
the  rectangle.  Distinguish  the  known  from  the  required. 
Review  method  of  discovering  solutions  to  originals.] 

If  you  fail,  see  the  "hint"  given  below. 

^Hint. — (2)  How  do  you  find  a  mea?i  proportional  to  two 
given  lines?  In  answering  this  question,  try  to  see  what  you 
have  given  which  can  be  used  as  data.  Of  what  two  lines  is 
A  B  the  sum?  What  is  the  mean  proportion  of  these  lines  ? 
{Ans.  Side  of  7n)  Can  you  construct  a  rt.  A  oi  which  A  B  is 
the  hypotenuse?  Construct  a  semicircle  on  A  B,  Draw  a 
line  parallel  to  A  B  (distance  equal  to  a  side  oim).  In  how 
many  points  will  this  parallel,  produced  if  necessary,  cut  the 
semi-circumference?  Draw  a  perpendicular  to  diameter  from 
either  of  these  points.] 

State  Prop.  XVIII. 

Discuss  this  problem :     When  is  it  impossible  ?  etc. 

Exercises. 

288.  The  sum  of  two  numbers  is  13;  their  product  is  36. 
Required  the  numbers. 

(1)     Solve  by  Algebra. 
{I)     Solve  by  Geometry. 

289.  Given  A  A  B  C  and  m  n  the  base  of  an  isosceles  A 
equivalent  to  A  B  C.  To  construct  the  isosceles  i\.  Is  this 
ever  impossible  ? 

290.  Given  the  rectangle  A  B  C  D,  the  Z  «.  and  the 
line  m  n.  To  construct  a  A  equivalent  to  A  B  C  D  having 
base  equal  to  vm,  and  adjacent  angle  equal  to  Z   A. 

291.  Upon  a  given  line  draw  an  isosceles  A  equivalent 
to  the  sum  of  a  given  A  and  a  given  quadrilateral. 


194 


PLANE  GEOMETRY,  BOOK  IV. 


285. 

Proposition  XIX. — Problkm. 
Given  the  square  m  and  the  line  A  B. 
Construct  a  rectangle  equivalent  to  m,  the  differ- 
ence of  whose  base  and  attitude  equals  A  B. 

[Hint. — What  proposition  for  finding  the  mean  propor- 
tional of  two  given  lines  did  you  use  in  §  284? 

Think  of  another  proposition  in  which  you  have  the  mean 
proportional  of  two  given  lines. 

Note  that  in  §  284  you  have  given  the  S7im  of  two  lines, 
while  in  §  285  you  have  given  the  — .  Try  hard  to  finish 
without  consulting  the  figure  given  below  or  reading  further. 


Construct  a  circle  having  diameter  A  B. 

Construct  a  tangent,  one  end  terminating  at  the  circum- 
ference, having  a  length  equal  to  a  side  o^  m.  Through  the 
extremity  without  the  circumference  draw  a  secant  passing 
through  the  center.  State  the  proposition  you  now  have. 
Find  what  the  square  m  equals,] 

State  Prop.  XIX. 


ExKRCiSE. 

292.     The   difference   of  two   numbers   is  10  and  their 
product  is  56.     Required  the  numbers. 

(1)  Solve  by  Algebra. 

(2)  Solve  by  Geometry. 

[Hint, — Take   any   unit   of  length.     Construct  the  line, 


AREAS. 


195 


286. 

Proposition  XX.—  Problem. 
Given  the  square  m  and  the  unequal  lines  p  and  q. 

It  is  desired  to  construct  a  square  that  shall  have 
the  same  ratio  to  m  as  that  ofp  to  q. 

Try  to  solve  without  consulting  "hint."  It  may  be  quite 
difficult,  but  try  repeatedly  by  using  all  you  have  learned 
about  methods  of  discovering  the  solution  of  original  prolems. 

[Hi7it. — Study  §  222.  Note  that  here  we  have  the  squares 
of  lines  proportional  to  lines.  Try  to  apply  this  in  the  above 
problem.] 


In  the  above  figure  p  and    q  are  proportional  to  what 
squares  ?     But  does  the  side  of  one  of  them  equal  the  side  of 

8 

m  ?     How  can  we  get  other  squares  having  the  ratio  of  A  C 

to  C  B  ?  How  can  we  get  two  squares,  one  of  which  equals  w? 
If  still  unable  to  solve,  cut  off  C  A.  C  E  a  distance  = 
a  side  of  w.  Then  through  E  draw  a  line  parallel  to  A  B, 
cutting  C  B  and  F.  Then  C  F  is  the  side  of  the  required 
square.  Discuss  the  above  problem.  Is  it  ever  impossible  ? 
Solve  when  p  L  q  \  when  p  ^=  q.  . 


Exercise. 

293.  If  area  of  w  =  100,  />  =  7,  and  ^  =  5,  find  the 
area  of  the  required  square.  Test  the  accuracy  of  your  answer 
by  a  drawing. 


196  .  PLANE  GEOMETRY,  BOOK  IV. 

287. 

Proposition  XXI. — Probi^em. 

Given  any  polygon,  A  B  C  D  K,  and  the  unequal  lines 
m  and  7i. 

Construct  a  polygon  similar  to  A  B  C  D  E  and 
having  the  ratio  to  it  of  m  to  n. 

If  you  fail,  see  "hint"  below. 

\Hint. — How  do  similar  polygons  vary?  Can  you  con- 
struct a  square  whose  ratio  to  the  square  of  a  side  of  A  B  C  D  E 
shall  equal  m  :  nT\ 

Discuss  the  problem. 

Exercises. 

294.  If  A  B  =  12  ;;?  =  15,  and  n  =  10,  find  A'  B'  side 
of  the  required  polygon. 

295.  Construct  a  A  equivalent  to  a  given  polygon  hav- 
ing given  the  base  and  median  to  the  base  of  the  A- 

288. 

Proposition  XXII. — Probi^em. 
Given  two  dissimilar  polygons,  A  and  B. 
//  is  required  to  construct  a  polygon  similar  to  A 
and  equivalent  to  B. 

[fii?u. — (1)     Assume  the  problem  solved. 
Then  we  have  given  dissimilar  polygons  A  and  B,  and  the 
required  polygon  X,  similar  to  A  and  equivalent  to  B. 


I^et  a-=^  2.  side  of  A,  and  x  =  2i  side  of  X,  the  required 
polygon. 


EXERCISES.  197 

Review  the  propositions  about  similar  polygons.  Form 
proportions. 

You  wish  to  discover  a  method  to  find  the  length  of  what 
line?         (With  it  known  you  can  construct  X.)] 

[///«/.— (2)     A:X  =  a^:;»;^     [.?] 

How  many  unknown  terms  in  the  above  proportion  ? 

Put  for  one  of  the  unknown  terms  a  known  term. 

Are  all  the  magnitudes  in  the  above  proportion  of  the 
same  kind? 

Get  another  proportion  from  this,  in  which  the  magni- 
tudes are  not  areas.     What  kind  of  magnitude  is  A  ? 

What  does  the  v^A  mean  f     Can  you  construct  it?] 

296.  Construct  a  rhombus  equivalent  to  a  given  rhomboid 

(1)  Having  one  diagonal  equal  to  the  short  diagonal 
pf  the  rhomboid; 

(2)  Having  one  diagonal  equal  to  the  long  diagoual 
of  the  rhomboid, 

297.  How  does  the  radius  of  an  inscribed  circle  in  an 
equilateral  A  compare  with  the  radius  of  the  circumscribed 
circle  ? 

298.  Construct  a  A  similar  to  one  of  two  given  dissimilar 
As  and  equivalent  to  their  difference.     Discuss. 

299.  Construct  an  isosceles  A  equivalent  to  a  given  A> 
having  given  one  of  the  equal  sides.     Discuss. 

300.  Draw  a  line  parallel  to  a  side  of  an  equilateral  A 
which  will  divide  it  into  two  equivalent  parts. 

\Hi7it. — (1)  Suppose  the  problem  solved,  and  then  compare 
the  area  of  the  A  cut  off  with  that  of  the  original  A-  How 
do  these  As  vary?     Form  the  proportion,  etc.] 


198  PLANE  GEOMETRY,  BOOK  IV. 

If  you  still  fail,  consult  figure  and  further  "hint." 


[ffhit., — (2)    Let  T  =  large  A  and  one  of  its  sides  =  «, 
T'  =  small  A  cut  off,  and  side  =  x. 
T:  T'  =  a'  :  j»;\    [?]     But  T  =  2  T ;  [  ?  ] 
.  • .  2  :  1  =  a^  :  x'^;    (What  is  the  unit  of  measure  here?) 
And  V^  :  1  =>  :  x.    (What  is  the  unit  of  measure  here?) 
Construct  the  ^2  where  the  unit  is  given.     Construct  the 
converse. 

301.  Draw  a  line  parallel  to  the  base  of  a  given  A  divid- 
ing it  into  two  equivalent  parts. 

302.  Draw  a  line  through  a  given  point  in  the  side  of  a 
,\'  dividing  it  into  two  equivalent  parts. 

[Hint. — Draw  a  scalene  A  and  solve,  fixing  the  point  in 
different  positions  in  the  sides. 


When  P  is  joined  to  opposite  vertex,  is  the  A  divided  into 
equivalent  As?  Why  not?  Where  must  P  be  in  order  that 
the  As  shall  be  equivalent?  Fix  this  point.  Call  it  m,  }oin 
P  and  7n  to  opposite  vertex.  The  line  from  m  cuts  off  half. 
How  much  does  the  line  through  P  lack  of  cutting  off  half? 
How  then  can  you  draw  a  line  through  P  which  will  cut  off 
in  addition  to  A  already  cut  off  a  A  equivalent  to  A  lacking?] 

303.  Draw  a  line  through  any  point  in  the  side  of  a  par- 
allelogram dividing  it  into  two  equivalent  parts.  When  will 
the  parts  be  As?   trapezoids? 


AREAS. 


199 


Moulding  of  Polygons. 

§283  may  be  stated  as  follows:  Mould  a  polygon  into  an 
equivalent  A-  Note  the  figure  carefully.  A  AC  F  is  equiv- 
alent to  A  A  C  B.  Why  ?  In  A  A  C  B  consider  A  C  the  base 
and  let  the  vertex  B  be  moved  parallel  to  the  base  A  C.  Sup- 
pose we  start  to  move  B  toward  F,  but  stop  at  intermediate 
points,  X,  y,  z,  etc.     Why  are  As  B  A  C,  X  A  C,  jj/  A  C,  2-  AC 

and  F  A  C  equivalent  ?     Are  any  of  these 

As  isosceles?     right?     equilateral? 

304.  I.  Redraw  ABAC  (§283)  and  mould  it  into 
(1)  an  isosceles  A  I  ('^)  a  right  A  I  (^)  a  different  isosceles  A 
from  (1).  What  is  this  base  here?  (-t)  Still  a  different  isosceles 
A  from  (1)  or  3.     What  is  now  the  base  ? 

II.  Now  redraw  ABAC  and  take  A  B  the  base. 
What  is  the  vertex?  Mould  B  A  C  into  the  following  As, 
using  A  B  the  base:  (1)  isosceles;  (2)  right.  Can  you  mould 
it  into  other  isosceles  As  while  A  B  is  considered  the  base? 
Why? 

III.  Answer  questions  as  asked  in  II.  using  B  C  the  base. 

IV.  Discuss  the  possibilities  of  moulding  a  scalene  A 
into  different  shaped  equivalent  isosceles  As. 

305.  Divide  any  quadrilateral  into  two  equivalent  parts 
by  drawing  a  line  through  any  given  point  in  any  one  of  its 
sides. 

{Hint. — Mould  the  quadrilateral  into  a  A-]     Discuss. 
308.     Draw   two  scalene  As   having   unequal  altitudes. 
Draw  an  isosceles  A  equivalent  to  their  sum. 


\_Hint. — (1)    Mould  each  A  into  a  right  l\.     Raise  the  al- 
titude of  T'  until  its  altitude  equals  that  of  T.     T  is  equivalent 


200  PLANE  GEOMETRY,  BOOK  IV. 

to  A  B  C.  [?]  T  is  equivalent  to  M  N  O.  [ ? ]  To  solve: 
(2)  Draw  M  R  and  then  draw  through  O  a  line  parallel  to  M  R, 
cutting  M  N  at  S.  Draw  S  R.  A  S  R  N  is  equivalent  to  A 
M  N  O,  which  is  equivalent  to  T'.     Why  ? 

(3)  Now  add  the  As  by  adding  their  bases  and  mould 
into  the  required  isosceles  A- 

307,     Draw  three  scalene  As  of  unequal  altitudes,  and 

(1)  Mould  into  a  rt.  A  equivalent  to  their  sum ;  (2)  Draw  a 
rectangle  equivalent  to  their  sum;  (3)  Draw  a  square  equiv- 
alent to  their  sum. 

808.  (1)  Draw  a  quadrilateral  and  mould  it  into  an 
equivalent  z\.  Now  mould  the  A  into  an  equivalent 
quadrilateral.  Discuss  the  data  necessary  to  mould 
the  A  into  the  original  quadrilateral. 

(2)     Draw  a  square  ^  as  large  as  a  given  square. 

309.  Draw^  an  irregular  pentagon  and  then  draw  a  square 
having  |  the  area  of  the  pentagon. 

310.  Mould  a  scalene   A   v^)  into  a  "kite"  trapezium; 

(2)  into  an  "arrow"  trapezium. 

311.  Construct  a  A  similar  to  a  given  A  and  having 
twice  the  area. 

312.  Draw  a  circle  having  twice  the  area  of  a  given 
circle. 

313.  Draw  a  square  having  |  the  area  of  a  given  square. 

314.  Draw  a  circle  having  half  the  area  of  a  given  circle. 

315.  Draw  an  irregular  polygon ;  then  draw  a  similar 
polygon  having  ^  the  area. 

316.  Draw  a  "kite"  trapezium  and  mould  it  into  an  equiv- 
alent A- 

317.  Draw  an  irregular  polygon  having  two  re-entrant 
Z  s.     Mould  it  into  an  equivalent  square. 

318.  Divide  a  given  line  into  2 J  equal  parts. 

319.  Draw  two  similar  but  unequal  rectangles.  Then 
draw  a  rectangle  similar  and  equivalent  to  their  sum. 


EXERCISES.  201 

330.     Divide  a  given  line  into  3  equal  parts,  not  using 

321.  Draw  a  line  through  the  vertex  of  a  given  A  so 
that  the  A  will  be  divided  into  two  As  which  shall  have  the 
ratio  2:3. 

322.  Construct  a  right  isosceles  A  equivalent  to  a  given 
square. 

323.  Find  the  locus  of  the  middle  points  of  all  the 
chords  in  a  given  circle  which  can  be  drawn  through  a  given 
point  (1)  in  the  circumference,  (2)  within  the  circle,  (3)  with- 
out the  circle. 

324.  Given  the  line  A  B  and  the  angle  a.  Construct 
largest  possible  A  that  shall  have  A  B  for  the  base,  an  Z  « 
for  the  vertical  angle. 

325.  From  a  point  without  a  circle  draw  a  secant  so  that 
the  intercepted  chord  shall  subtend  \  of  the  circumference. 

326.  From  a  point  without  a  circle  draw  a  secant  so  that 
the  internal  segment  and  the  external  segment  shall  be  equal. 
Discuss  Ex.  326. 

327.  Construct  a  triangle  having  given — 

(1)  The  base,  altitude,  and  the  median  to  the  base. 
Discuss, 

(2)  The  angles  and  one  median.     Discuss. 

(3)  One  side,  an  Z  adjacent  to  that  side,  and  the 
sum  or  the  difiference  of  the  other  two  sides. 
Discuss. 

(4)  The  perimeter,  one  Z  ,  and  the  altitude  drawn 
from  the  vertex  of  this  Z  .     Discuss. 

(5)  The  radius  of  the  circumscribed  circle,  and  the 
angles.     Discuss. 


202  PLANE  GEOMETRY,  BOOK  V. 


BOOK    V. 

REGULAR  POLYGONS. 

Measurement  of  the  Circi^e, 

328. 

Recall  your  definition  of  a  regular  polygon.     See  §  104. 
The  term  regular  polygon  means  a  convex  polygon  unless 
otherwivSe  stated. 

What  would  you  call  an  equilateral  triangle?     a  square? 

329. 


Regular  Convex  Pentagon.       Regular  Concave  or  Cross  Polygon. 

The  subject  of  the  regularity  of  polygons  may  be  looked 
at  from  the  standpoint  of  symmetry.  By  symmetry  in  Geom- 
try  we  mean  that  if  a  figure  be  turned  half  way  round  on  a 
point  as  a  pivot,  each  part  of  the  figure  will  occupy  the  same 
space  previously  occupied  by  another  part.  If  the  figure,  on 
being  turned  halfway  round,  occupies  its  original  position,  it 
is  said  to  have  two-fold  symmetry. 

If  an  equilateral  triangle  be  revolved  about  its  center 
one-third  of  360°,  what  will  be  its  second  position?  Suppose 
it  is  turned  two-thirds  of  360°  about  the  center,  what  will  be 
its  third  position? 


SYMMETRY. 


203 


Discuss  revolving  a  square  about  its  center. 

Is  a  right  triangle  symmetrical  with  regard  to  its  center? 
an  isosceles  triangle  ? 

If  a  figure  be  turned  one-third  of  a  revolution  and  it 
occupies  its  original  position,  the  figure  is  said  to  have  three- 
fold symmetry. 

What  is  four- fold  symmetry?    five-fold  symmetry? 

What  is  the  symmetry  with  regard  to  a  point  illustrated 
by  the  following  figures?  Make  figures  illustrating  other 
forms  of  S3^mmetry. 

For  examples,  observe  wall  paper  and  other  decorations. 


Fig.  1 


Fig.  2. 


Fig.  3. 


Fig.  5. 


Fig.  6. 


304  PLANE  GEOMETRY,  BOOK  V. 

330. 

From  the  standpoint  of  symmetry,  polygons  that  are  sym- 
metrical are  regular.  Thus,  a  triangle  is  regular  if  it  has 
three-fold  symmetry.  A  heptagon  is  regular  if  it  has  seven- 
fold symmetry. 

A  polygon  of  n  sides  is  regular  if  it  has  «-fold  symmetry. 

331. 

By  means  of  revolution  show  that  a  symmetrical  octagon 
has  (1)  its  sides  equal,  (2)  its  angles  equal;  (3)  that  a  circle  may 
be  circumscribed  about  a  regular  polygon  having  the  same 
center  as  the  polygon;  (4)  that  with  the  center  of  the  polygon 
for  center  a  circle  may  be  inscribed  within  it. 

From  the  special  case  just  given,  can  you  prove  a  general 
truth?    State  it. 

Notice  that  we  prove  by  symmetry  (1)  and  (2),  which  are 
given  as  a  definition  in  §  104. 

332. 

Into  how  many  isosceles  triangles  may  a  regular  triangle 
be  divided  by  joining  the  center  to  the  vertices  ?  a  regular 
quadrilateral?  a  regular  pentagon?  a  regular  hexagon  ? 

Proposition  I. 

Can  you  show  that  a  regular  polygon,  P,  of  n 
sides,  may  be  divided  into  ;/  isosceles  triangles  ? 
Write  Prop.  I. 

333. 

Cor.  I.  What  do  the  bisectors  of  any  two 
Z  s  of  a  regular  polygon  determine  by  their  inter- 
section? 


REGULAR  POLYGONS.  205 

334. 

Cor,  II.  Appoint  is  equidistant  from  all  the 
vertices  of  a  regular  polygon.  How  is  it  related  to 
the  sides  of  the  polygon?     Proof. 

335. 

Cor,  III,  Show  that  a  O  may  be  circumscribed 
about  or  inscribed  within  a  regular  polygon  and 
both  Os  have  the  same  center. 

The  center  in  Cor.  III.  is  called  the  center  of  a  regular 
polygon  and  the  radius  of  the  circumscribing  circle  is  called 
the  radius  ol  a  regular  polygon;  the  radius  of  the  inscribed 
O  is  called  the  apothem  of  a  regular  polygon.  Draw  figure 
and  fix  these  terms  in  mind.  What  is  meant  by  the  Z  at  the 
center  of  a  regular  polygon? 

336. 

Cor,  IV.  If  a  regular  polygon  have  n  sides^ 
show  that  each  Z  at  the  center  =  4  rt.  Is  -^  n. 

337. 

Proposition  II. 

Given:  A  B  C  D  E  F,  an  equilateral  polygon  in- 
scribed in  a  circle.     Can  you  prove  it  to  be  regular? 


338. 

Cor,  /.  If  a  circumference  be  divided  into  n 
equal  arcs,  {n  >  2),  what  will  the  chords  of  these 
arcs  form? 


306  PLANE  GEOMETRY,  BOOK  V. 

339. 

Cor.  II,  If  the  arcs  subtended  by  a  regular 
polygon  of  n  sides  be  bisected,  what  will  the  chords, 
of  these  arcs  form? 

340. 

Proposition  III. 


Suppose  A  B  C  D  E  and  A'  B'  C  D'  E'  are  regular  poly- 
gons of  the  same  number  of  sides.  (1)  Compare  the  sum  of  all 
the  Z  s  in  P  and  P'.  (2)  Compare  Z  A  and  Z  A',  Z  B  and  Z  B'. 
Why  does  Z  A  =  Z  A?  Z  B  =  Z  B'?  Compare  A  B,  B  C, 
C  D,  etc.     Compare  A'B',  B'C,  CD',  etc. 

Can  you  show  that  P  and  P'  are  similar  ? 

Call  this  Prop.  III. 

341. 

Proposition  IV. 

Prove  that  regular  polygons  of  the  same  num- 
ber of  sides  may  be  divided  into  the  same  number 
of  similar  As,  similarly  placed. 


REGULAR  POLYGONS. 
342. 


207 


Proposition  V. 


In  A — D  and  A' — D'  we  have  two  regular  polygons  of 
the  same  number  of  sides.  Let  P  and  P'  be  the  perimeters 
and  S  and  S'  be  the  areas  of  the  figures.  Let  O  and  O'  be 
the  centers.  (How  find  them  ?)  Join  O  A,  O  B,  O'  A',  O'  B', 
and  draw  the  Is  O  F  and  O'  F'  to  A  B  and  A'  B'  respectively. 
Call  O  A,  R;  O'  A',  R';  O  F,  r;  O'  F',  /.  Compare  Z  s  A  O  B 
and  A'  O'  B'.  Can  you  compare  the  ratio  of  O  A  and 
O'  A'  with  ratio  of  O  B  and  O'  B'  ?  What  conclusion  concern- 
ing the  As?  Compare  the  ratio  of  A  B  and  A'  B'  with  the 
ratio  of  O  F  and  O'  F?  Substitute  for  O  B,  O'  B',  O  F,  O'  F 
their  values  in  the  last  comparison. 

Can  you  now  snow  that  p,  =  ^,  =  ^ .'' 


Show  that  g,  =  7^ 


(Rr 


What  are  R,  R',  r,  /  ? 

Can  you  state  the  proposition  and  prove  ? 

Call  this  Prop.  V. 


208  PLANE  GEOMETRY,  BOOK  V. 

343. 

Proposition  VI. 

Show  how  to  find  the  area  of  any  regular  polygon. 
Word  the  conclusion  Prop.  VI. 

344. 

Proposition  VII. 

Show  how  to  inscribe  a  square  in  a  given  ©. 

Call  this  Prop.  VII.  "     ' 

345. 

Proposition  VIII. 

Can  you  prove  how  to  inscribe  a  regular  hexagon  in  a 
circle  ? 

KXERCISKS. 

328.  In  how  many  ways  can  you  construct  an  equilat 
eral  triangle  ? 

329.  If  r  =:  radius  of  the  O,  and  s  the  side  of  the  in- 
scribed equilateral  A,  show  that  s  =  r  \JS. 

330.  The  distance  from  the  center  of  an  inscribed  equi- 

y 

lateral  A  to  a  side  is  ^ 

381.  Inscribe  an  equilateral  A  and  a  regular  hexagon  in 
the  same  ©.     Compare  their  areas. 

332.  Circumscribe  an  equilateral  A  about  the  Q  i^i  Ex- 
ercise 331. ,  Compare  the  hexagon  with  the  2  As.  State  your 
conclusion  in  neat  form. 


REGULAR  POLYGONS.  209 

346. 

Proposition  IX. — Probi^em. 


M               F              K, 

•n^^'^==^  ,-' 

In  this  figure  let  A'  B'  be  one  side  of  a  regular  circum- 
scribed polygon  of  n  sides.  Call  its  perimeter  P,  and  let  O  be 
the  center  of  the  polygon  and  circle.  Draw  O  A',  O  B'  cut- 
ting the  arc  at  A  and  B.  Join  A  and  B.  Prove  A  B  ||  to  A'  B'. 
Can  you  show  that  A  B  is  the  side  of  a  regular  inscribed 
polygon  of  n  sides?     Call  its  perimeter  p. 

Join  A  F,  B  F  and  draw  tangents  at  A  and  B.  How  often 
is  A  B  used  in  the  regular  inscribed  polygon?  Show  that 
A  F  is  the  side  of  a  regular  inscribed  polygon.  How  many 
sides  ?  Can  you  show  that  M  N  is  the  side  of  a  regular  cir. 
cumscribed  polygon?  How  many  sides  to  the  polygon  of 
which  M  N  is  a  side?  Call  the  perimeter  whose  side  is  M  N, 
P',  and  the  regular  polygon  whose  side  is  A  F,  p\     Show  that 

A'  B'  =  ~.     Write  the  values  of  A  B,  M  N,  A  F.    Join  O  M, 

O  F.     Prove  that  M  O  bisects  Z  A'  O  F.     Show  that  ^r^= 

M  F 

O  A' 

7;-^=r.     What  are  O  A'  and  O  F  of  the  polygons?     How  are 

the  radii  of  regular  polygons  of  the  same  number  of  sides 

P      A'  M 
related?    (§  342.)     What  axiom  shows  this,  -  =  r— ^? 

Rewrite  this  equation  by  composition.  What  is  the  sum 
of  A'  M  and  M  F?    How  is  M  F  related  to  M  N? 


210  PLANE  GEOMETRY,  BOOK  V. 

p  _1_  V)        1    A  '  "P' 

Do  you  see  that — ur=|-— — -?  Substitute  the  values  of 

/>         4  M  N 

A'  B'  and  M  N.     Can  you  now  show  (1)  F  =        ^  ? 

What  is  P'  ?     Express  this  equation  in  words. 
Can  you  prove  As  A  B  F  and  A  M  F  similar?     Show 
that  ATf'  =  a  B  -  4  M  N.     Substitute  the  values  of  A  F, 


A  B,  M  N,  and  show  that  (2)  p'  =  \!p  -  P'. 

By  means  of  (1)  and  (2)  the  perimeter  of  regular  inscribed 
and  circumscribed  polygons  of  double  the  number  of  sides 
may  be  found,  for  any  values  of  P  and  p. 

Call  the  above  Prop.  IX.     Write  a  statement  of  it. 

EXERCISKS. 

333.     The  side  of  an  inscribed  sq.  z=zr  \I2 

384.  If  s  denotes  the  side,  R  =  radius,  ?  =  apothem, 
and  S  =  area,  and  the  radius  of  the  Q  =  1,  prove  that: 

(a)     In  a  regular  inscribed  octagon  s=  \/2 \/%  r  •=. 

J  v/2  4-  V~2,  S  =  2  V2: 

{b)  In  a  regular  circumscribed  octagon  ^  =  2  v/2  —  2, 
R  =  V4  —  2\1%   S  ==  8  v/2  —  8.  

{c)  In  a  regular  inscribed  dodecagon  .?  =  V(2  —  v^s)"^ 
^=4  V2  -f   Vl,  S  =  3. 

{d)    In  a  regular  circumscribed  dodecagon 
^  =  4  —  2  v/3,  R  =V8  —  4  v's,  S  =  24  —  12  \/3. 

335.     In  a  given  sector  whose  Z  at  the  center  =  90^ 

R^ 
inscribe  a  square.    Prove  the  area  =  -«— . 


REGULAR  POLYGONS.  211 

347. 

Proposition   X. 

Problem.      To  inscribe  a  regular  decagon  in    a 
circle. 


Sug.  1.  Divide  the  radius  C  A  into  mean  and  extreme 
ratio  and  draw  the  chord  A  B  equal  to  C  M.  Join  M  with  B, 
C  with  B.  When  is  a  line  divided  into  mean  and  extreme 
ratio  ?     Write  proportion. 

Stig.  2.  In  place  of  C  M  put  its  equal  A  B  in  the  pro- 
portion above.  Can  you  compare  As  CAB  and  M  A  B 
through  the  last  proportion  ?     Quote  proof. 

Su^.  3.  Compare  B  A  and  B  M,  B  M  and  C  M.  Com- 
pare Z  A  B  M  with  ZsAC  BandMBC,  ZABC  with 
ZACB,  Z  BAC  with  Z  A  C  B.  How  many  times  the 
angle  A  C  B  is  the  sum  of  the  angles  of  the  triangle  A  C  B? 

What  is  the  sum  of  the  Z  s  in  the  A  ? 

Show  that  Z  A  C  B  =  36°. 

A  B  is  the  chord  of  what  angle  at  the  center?  How 
many  times  may  it  be  applied  to  the  circumference  ? 

Give  the  method  of  inscribing  a  regular  decagon  in  a 
circle. 

348. 

Cor.  I.     How  inscribe  a  regular  pentagon  ? 


212  PLANE  GEOMETRY,  BOOK  V. 

349. 

Cor,  II.  At  a  given  point  in  the  circumference  make  a 
chord  equal  to  the  side  of  the  inscribed  decagon;  from,  the 
same  point  in  the  same  direction  along  the  circumference 
make  a  chord  equal  to  the  side  of  a  hexagon.  What  part  of  a 
circumference  is  the  difference  between  the  extremities  of 
these  2  chords?  The  chord  of  the  difiference  of  these  arcs  is 
the  side  of  what  regular  inscribed  polygon? 

5.  Name  the  series  of  regular  polygons  that  we  have 
learned  to  inscribe.  Read  a  biography  of  the  great  mathe- 
matician Gauss. 

Exercises. 

336.  In   the  figure  under  §  347  let  x  — C  M.     Can  you 

show  that  A  B  =   — ^^ — ^ ?     Word  this  equation. 

337.  When  the  side  of  a  regular  decagon  is  10  feet,  cal 
culate  the  radius  of  the  decagon  and  test  the  accuracy  of  your 
answer  by  drawing  to  scale  of  .1  of  an  inch  to  the  foot. 

338.  Prove  that  the  angle  at  the  center  of  a  regular 
polygon  is  the  supplement  of  the  angle  of  the  polygon. 

350. 

Proposition  XI. — Probi^km. 
Draw  a  regular  inscribed  polygon.     Bisect  the  arcs  sub- 
tended by  the  equal  chords,  and  at  these  points  of  bisection 
draw  tangents. 

Prove  that  these  tangents  form  a  regular  circum- 
scri bed  polygon. 

Call  this  Prop.  XI. 

351. 

Cor.  By  joining  certain  points  in  the  last  fig- 
ure, show  that  you  can  make  a  regular  inscribed 
polygon  of  twice  the   number  of  sides  of  the  origi- 


REGULAR  POLYGONS.  213 

nal;  and,  by  joining  certain  other  points,  a  regular 
circumscribed  polygon  of  double  the  number  of  sides 
of  the  original  circumscribed  polygon. 

352. 

What  regular  circumscribed  polygons  may  be  constructed? 

EXERCISK. 

339.     Prove   the  diagonals   of  a  regular    pentagon   are 
equal. 

353. 

Proposition  XII. 


Let  the  irregular  curve  M  N  D  K . . . .  envelop  the  circum- 
ference MRS.  Draw  A  E  tangent  at  T.  Compare  A  E  with 
A  H  E.  Tangent  E  K  with  ELK.  Compare  A E L  K D  M 
with  A  H  E  L  K  D  M.  What  is  the  shortest  enveloping  line 
of  the  surface  within  M  R  T  S?  How  does  the  circumference 
compare  with  any  enveloping  line  ?    State  Prop.  XII. 

354. 

Cor.  How  does  the  circumference  of  a  ©  compare 
with  the  perimeter  of  any  regular  circumscribed  polygon? 
with  any  regular  inscribed  polygon? 


214  PLANE  GEOMETRY,  BOOK  V. 

355. 

Proposition  XIII. 


I.  lyet  P  and/>  be  the  perimeters  of  the  regular  circum- 
scribed and  inscribed  polygons  of  the  same  number  of  sides, 
and  S  and  s  the  areas,  and  C  the  circumference  of  the  Q-  I^et 
A'  B'  be  a  side  of  the  polygon  whose  perimeter  is  P.  Draw 
OA',OB',  CBandOFlA'B'.    What  is  CB?    [§346.]  Prove 

JP_0  A' 
that  (1)  p  ~Q^'     By  division  what  does  (1)  equal?     Show 

that  P—  p  =^  (O  A'— O  F). 

In  A  O  A'  F  show  that  O  A'— O  F  <  A'  F.  What  does 
A' F  J=  as  you  continue  to  increase  the  number  of  sides? 
What  will  O  A'  -  O  F  ^  ?  What  will  C  F'  =  ?  What  will 
P  —  C  and  C  —  />  =L  ?  What  do  P  and  /  =l  ?  Express  this  in 
the  form  of  a  proposition. 

II.  Let  us  find  the  limit  of  the  variables  S  and  s. 


S       OA'  S— 5      A'  F 

^^=  ^^^.    Why?     Prove  that* -7-  =  ^^=^  and  that 

^       O  F  OF 

S— J  =        A  •  A'  F  .    Increase  the  number  of  sides  indefinitely 
OF 


What  does  A' F  =l  ?  What  does  S  — J  J=?  How  does  the 
area  of  the  O  compare  with  S  ?  with  s  ?  Call  area  O  A. 
What  does  S  —  A=?    A  —  ^  =  ^    What  are  the  variables? 


REGULAR  POLYGONS.  215 

What  then  do  S  and  ^  =^  ?     Make  a  general  statement  of  this 
proposition,  including  both  parts. 

356. 

Cor.  What  limit  has  the  radius  of  the  circum- 
scribed polygon  and  the  apothem  of  the  inscribed 
polygon  ? 

Exercises. 

340.  Can  you  prove  that  the  diagonals  of  a  regular  pen- 
tagon cut  each  other  in  extreme  and  mean  ratio  ? 

341.  In  the  regular  pentagon  ABC  D  K  let  AD  and 
B  E  cut  each  other  at  P ;  prove  that  AP:AE::AE:AD. 

342.  Construct  a  regular  pentagon  equivalent  to  the 
sum  of  two  given  regular  pentagons. 

343.  Let  d  =  side  of  an  inscribed  regular  decagon,  p  = 
that  of  an  inscribed  regular  pentagon,  r  =  radius  of  the  O- 
Prove  that />'  =  d^  ^  r\ 

344.  Given  the  side  of  a  regular  pentagon  to  construct 
the  pentagon. 

345.  Given  the  side  of  a  regular  hexagon  to  construct 
the  hexagon. 

346.  Construct  the  regular  hexagon  A  B  C  D  E  F.  Draw 
the  diagonals  D  F,  E  A,  F  B.  What  new  figure  is  found? 
Compare  with  the  original  figure. 

357. 

Definition:  In  unequal  circles,  similar  arcs,  sectors  or 
segments  are  those  which  have  equal  angles  at  the  center. 


216  PLANE  GEOMETRY,  BOOK  V. 

358. 
Proposition  XIV. 


'1  2 

In  1  and  2  let  R,  ^,  and  C,  c,  and  S,  s  denote  respectively 
tlie  radii,  circumferences,  and  surfaces  of  the  ©s.  Inscribe  the 
2  regular  polygons  of  the  same  number  of  sides,  letting  P,  p 
and  A,  a  denote  respectively  the  perimeters  and  areas.  Com- 
pare the  perimeters  with  their  radii?  Write  the  relation. 
Compare  the  areas  with  their  radii.  Increase  the  number  of 
sides  indefinitely,  keeping  them  the  same  in  number.  Does 
the  ratio  written  above  ever  change  ?  What  does  P  =?  What 
does  />  =  ?  What  does  A  =  ?  a  ^  }  If  2  variables  are  in  a 
constant  ratio,  what  can  be  said  of  their  limits?  Prove  your 
answer. 

C     R  S     R 

Show  that  ~  =  —  and  that  -=  -r-    Express  this 

clearly. 

359. 


CD         S      D^ 
Cor.  I.    Prove  ^=-J  and  ~~  -j^*    Express  this 


d         ,s       d 


clearly. 


REGULAR  POLYGONS.  217 

360. 

Cor.  II.     (1)  Write  the  proportion  of  Cor.  I.  by 
alternation. 

Do  you  see  that  this  proportion  may  be  interpreted  as 

showing  that  the  ratio  of  the  circumference  of  a  ©  to  its 

C 
diameter  is  a  constant  quantity?     This  constant—  is  denoted 

by  the  Greek  letter  tt.  It  is  the  initial  letter  of  the  Greek 
word  for  circumference  {periphereia).  It  is  proved  by  methods 
in  higher  mathematics  that  n  is  incommensurable? 

(2)  Prove  that  C  =  7rD  =  27rR. 

(3)  Show  how  similar  sectors  are  related.^  (How 
are  two  ©s  related  ? 

Exercises. 

347.  Find  in   terms  of  the  radius  and  diameter  of  the 
circle  the  perimeter  of  a  regular  inscribed  hexagon. 

348.  Find  in  terms  of  the  radius  and  diameter  the  per- 
imeter of  a  regular  circumscribed  hexagon. 

349.  Solve  as  in  Exercises  347  for  the  perimeter    of   a 
regular  inscribed  dodecagon. 

350.  Solve  as  in   Exercises   348  for  the    perimeter  of  a 
regular  circumscribed  dodecagon. 

351.  Show  that  the  area  of  a  circle  is  four  times  the  area 
of  a  circle  described  on  the  radius  as  a  diameter. 

352.  How  does  the  square  inscribed  in  a  semicircle  com- 
pare with  the  area  of  the  circle? 


218  PLANE  GEOMETRY,  BOOK  V. 

361. 
Proposition  XV. 


Let  R,  C,  and  A  denote  the  radius,  circumference,  and 
area  of  ihe  Q.  Construct  a  polygon  of  71  sides.  Call  its  per. 
imeter  P,  and  apothem  R,  and  area  S.  Write  an  expression 
for  the  area  of  the  polygon.  As  the  number  of  sides  are  in- 
definitely increased,  what  does  S  =  .-*  what  does  P  =?  what 
does  ^  P  R  =?  Now  if  S  =  A  or  |  P  R  =^  A  and  J  P  R  = 
^  C  R,  then  how  are  A  and  J  C  R  related  ?  Review  each  step 
of  your  proposition.     Make  it  clear.     State  the  theorem. 

362. 

Cor,  I,  Substitute  the  value  of  C  in  terms  of 
R  and  show  that  A  =  ^^  R^ 


363. 

Cor.  11,  Show  that  the  area  of  a  sector  =  i  of 
the  product  of  the  arc  by  its  radius.  Write  the 
steps  of  your  proof. 


REGULAR  POLYGONS. 


219 


364. 

Taking  the  formulae  in  §346,  P'  =  t>    \^  ^^^  P'  "^  V^  x  P', 

and  calling  the  diameter  of  the  circle  1,  can  you  show  how  we 
may  approximate  the  ratio  of  the  diameter  of  a  circle  to  its 
circumference? 

From  the    above  formulae  the  following  table  has  been 
computed  : 


Perimeter 

Perimeter 

No.  Sides. 

of  Circumscribed 

of  Inscribed 

Polygon. 

Polygon. 

4 

4.00000 

2  82813 

8 

3.31371 

3.06147 

16 

3.18260 

3.12145 

32 

3.15172 

3.13655 

64 

3.14412 

3.14033 

128 

3.14222 

3.14128 

256 

3.14175 

3.14152 

512 

3.14163 

3  14157 

Notice  that  by  the  last  result  the  approximate  value  of 
the  ratio  of  the  circumference  to  the  diameter  is  3.141  (*),  cor- 
rect to  the  4th  decimal  place.     That  is,  tt  =  3.1416. 

Exercises. 

353.  An  oyster  can  is  4  inches  in  diameter  and  8  inches 
high.     How  many  square  inches  of  tin  are  required  to  make  it  ? 

354.  Find  the  length  of  an  arc  of  180°  in  a  circle  of 
radius  4. 

355.  A  circus  ring  contains  40  square  rods.  Find  its 
radius  and  circumference.     Call  tt,  3|. 

356.  The  apothem  of  a  hexagonal  paving-stone  is  18 
cm.     Find  the  area  of  its  circumscribing  O. 

357.  How  many  degrees  in  an  arc  whose  length  =  the 
length  of  the  radius  of  the  circle?  This  arc  is  called  a  radian 
and  is  one  of  the  units  for  measuring  circles. 


220 


PLANE  GEOMETRY,  BOOK  V. 


358.  A  cow  is  tethered  with  a  chain  15  m.  long;  the 
stake  is  driven  10  m.  from  a  straight  fence.  Over  how  much 
ground  can  the  cow  graze  ? 

359.  A  railroad  fence  meets  a  farmer's  fence  at  an  angle 
of  50°;  the  farmer  tethers  a  cow  between  the  fences  at  the 
corner  post;  the  chain  is  30  m.  long.  Over  how  much  ground 
can  the  cow  graze  ? 

860.  Construct  a  rt.  A,  circumscribe  a  circle  about  the 
A»  and  on  each  side,  about  the  rt.  Z  as  a  diameter,  describe  a 
semicircle  exterior  to  the  A-  Compare  the  sum  of  the  cres- 
cents with  the  area  of  the  A- 

Maxima  and  Minima  or  Pi^ane  Figures. 

365. 

When  thinking  of  qualities  of  the  same  kind,  that  which 
is  greatest  is  called  maximum  and  that  which  is  least  is  called 
minimum.  What  is  the  maximum  chord  in  a  Q  ?  What  is 
the  minimum  line  from  a  point  to  a  line  ? 

366. 

Proposition  I. 


Two   figures  are  isoperimetric  when   they    have    equal 
perimeters. 


MAXIMA  AND  MINIMA  OF    PLANE   FIGURES. 


221 


367. 


In  this  figure  let  the  2  As  ABC  and  A'  B  C  have  two 
sides  of  one  =  two  sides  of  the  other;  /.  ^.,  A  B  =  A'  B  and 
B  C  =  B  C.  A  A  B  C  is  a  right  A  having  right  Z  at  B.  Drop 
the  1  A'  D.  Compare  A'  B  and  A'  D.  Express  this  relation. 
How  does  A  B  compare  with  A'  D?  Suppose  we  draw  any- 
other  line  =  A  B,  as  A"  B.  Join  A"  to  C.  Draw  1  A"  E. 
Compare  A"  E  with  A  B.  Compare  the  As  A'  B  C  and  A"  B  C 
with  the  right  A- 

Of  all  As  having  2  sides  equal,  which  is  the 
maximum? 

368. 

Proposition  TI. 


Let  A  B  C  be  an  isosceles  A  and  A'  B  C  be  an  equivalent 
A  on  same  base.     Compare  their  perimeters. 

[///;//'.— Rroduce  B  A  to  D,  making  AD==AB=AC 
Join  D  and  C  and  A  A'.  Produce  A  A'  to  E.  Compare  alti- 
tudes of  As  A  B  C  and  A'  B  C.  Compare  A  A'  E  and  B  C,  E  D 
with  E  C,  A  E  with  D  C,  A'  D  with  A'  C] 

Can  you  show  that  the  perimeter  of  A  A  B  C  is 
less  than  the  perimeter  of  A  A'  B  C? 

Generalize  this  theorem. 


222 


PLANE  GEOMETRY,  BOOK  V. 

369. 


Cor,     Of  all  equivalent  As,  which  has  the  least 
perimeter? 


370. 


Proposition  III. 


Construct  A  A  B  C  with  A  B  =  A  C.  Draw  A  D  1  B  C 
and  make  A'  B  +  A'  C  ==  A  B  +  A  C.  Draw  A'  B  ||  B  C 
meeting  A  D,  or  A  D  produced,  in  E.  Join  E  B  and  E  C. 
Compare  As  E  B  C  and  A'  B  C.  Compare  A'  B  +  A'  C  with 
EB+EC,  AB-f  AC  with  E  B  +  E  C.  A  B  with  E  B, 
A  D  with  E  D. 

[Use  Doctrine  of  Exclusion.  Why  is  D  E  not  equal  to 
DA?     Why  not  greater  ?] 

Which  is  the  maximum  A.^ 

Write  a  general  statement  of  the  truth  proved. 


371 


Cor,     Of  all  isoperimetric  As,  which  has   the  greatest 


area 


LINES  AND   PLANES  IN  SPACE. 


223 


Exercises. 

361.  The  perimeter  of  a  maximum  A  is  7i  meters.  Find 
its  area. 

362.  Show  what  is  the  greatest  A  that  can  be  inscribed 
in  a  O. 

383  If  the  diagonals  of  a  parallelogram  are  given,  when 
is  its  area  a  maximum?  When,  if  ever,  may  the  maximum 
parallelogram  be  a  square? 

372. 

Proposition  IV. 


Let  A  B  C  D  E  be  a  plane  figure  bounded  by  the  convex 
arc  BED  and  the  concave  arc  BCD  with  the  staight  line  B  D 
joining  the  ends  of  the  concave  arc.  Show  that  A  B  C  D  E 
cannot  be  the  maximum  of  isoperimetric  figures. 

Siig.  Revolve  B  C  D  on  the  axis  B  D  till  it  comes  into 
the  plane  of  the  original  figure.  Compare  the  two  perimeters. 
Can  you  draw  any  conclusion  as  to  the  form  of  a  closed  figure 
of  given  perimeter  if  it  is  to  have  a  maximum  area? 


224  PLANE  GEOMETRY,  BOOK  V. 

373. 

Proposition  V. 


In  this  figure  let  the  curve  A  C  B  have  a  given  length. 
Join  the  ends  A  and  B  with  the  line  A  B.  Suppose  the  fig- 
ure to  be  a  maximum. 

Required  to  find  its  form. 

Take  any  point  P  on  the  line  and  join  P  and  A,  P  and  B. 
Call  the  segments  cut  off  by  P  A  and  P  B,  S^  and  S^,  and  the 
A  formed,  t.  Suppose  the  A  is  not  a  maximum,  what  do  you 
know  of  Z  A  P  B?  What  must  Z  P  be  that  the  /\  may  be  a 
maximum? 

Imagine  S^  and  Sg  hinged  at  P  as  a  pair  of  compasses 
with  unequal  legs.  Imagine  Z  P  >  a  right  Z ,  and  suppose 
the  point  A  slipped  along  A  B  to  A'  till  Z  A'  P'  B  is  a  right 
Z .  What  has  been  done  to  the  area  of  the  A  ?  Have  you 
changed  the  area  of  S^  and  S^?  Have  you  changed  the  area 
of  Si  +  /  +  S2,  or  the  figure  ABC?  Is  this  possible  if 
A  B  C  is  a  maximum? 

In  what  kind  of  a  curve  is  the  A  located? 

Why? 

Write  a  generalization  of  your  conclusion.  Call  it 
Prop.  V. 


MAXIMA  AND  MINIMA  OF    PLANE   FIGURES. 


225 


374. 

Proposition  VI. 


Given  the  convex  figure  A  C  B  D.  Let  A  and  B  be  2 
points  that  bisect  the  perimeter.  Can  you  show  that  if  the 
figure  is  a  maximum,  the  straight  line  A  B  bisects  the  area? 

Sug.  Suppose  S,  >  S,  and  then  revolve  Sj  on  A  B  into 
the  plane  of  the  paper.     What  would  follow? 

If  Sj  and  Sj  have 'maximum  areas,  what  must  their  forms 
be?     Why? 

What  is  A  B  of  the  maximum  figure  A  C  B  D? 

Generalize. 

375. 
Proposition  VII. 


Fig.  1.  Fig.  2.  Fig  3. 

Suppose  the  O  Figure  1  and  the  plane  Figure  2  to  have 
equal  areas  and  that  the  perimeter  of  Figure  2  equals  the  per- 
imeter of  circle  Figure  3.  Compare  the  areas  of  Figure  2  and 
Figure  3,  of  Figure  1  and  Figure  3;  the  circumferences  of  Fig- 
ure 1  and  Figure  3;  the  circumference  of  Figure  1  with  the 
perimeter  of  Figure  2.     Generalize. 


226 


PLANE  GEOMETRY,  BOOK  V. 


376. 

Proposition  VIII. 


Given  the  polygon  A  B  —  E  inscribed  within  the  circle 
A  B  —  K.  Let  A'  B^  —  E'  be  another  polygon  having  the 
sides  A'  B'  =r  A  B,  B'  C=B  C,  etc.  Construct  segments  A'  B', 
B'  C,  etc.,  =  to  the  corresponding  segments  on  sides  A  B, 
B  C,  etc. 

Compare  the  perimeter  of  the  circle  and  the 
curvilinear  figure.  Compare  their  areas.  Can  you 
now  show  that  the  plane  figure  A  B  —  E  >  A'  B'  — 
E'?     Write  theorem. 

377. 
Proposition  IX. 

Eet  A  B  C  D  E  be  the  maximum  polygon  having  a  given 
perimeter  and  n  sides.  Join  A  C.  Suppose,  if  possible,  A  B  and 
B  C  2inequal  and  let  A  B'  C  be  an  isos.  A>  A  B'  =  C  B'  and  iso- 

perimetric  with  ABC.    Compare  area  A  B'  C  D  E with 

the  original  figure.  Can  A  B'  C  D  E  ...  be  greater  than 
ABCDE  ....?  What  conclusion  can  you  draw  concern- 
ing A  B  and  B  C  ? 

Can  you  show  that  AB  =  AE=ED  =  DC? 
Is  this  polygon  inscriptible?  Give  reason  for  an- 
swer.    Is  it  regular?     Write  theorem. 


MAXIMA  AND  MINIMA  OF    PLANE   FIGURES. 

378. 

Proposition  X. 


227 


Let  A  B  C  D  be  a  square.  Take  any  point  Pon  side  A  D 
and  construct  /\  F  x  C  isoperimetric  with  P  D  C  and  having 
the  side  F  x  =  C  x. 

(1)  Compare  Z\s  P  D  C,  P  ^  C  (2)  Compare 
area  of  square  with  pentagon  A  B  Cr  P.  (3)  Suppose 
pentagon  were  made  regular  and  isoperimetric  with 
A  B  C  :f  P,  how  would  area  compare  with  irregu- 
lar pentagon  and  square?  (4)  Can  you  show  that  a 
regular  hexagon  >  a  regular  isoperimetric.  penta- 
gon ?       (5)    How   far  may    this  reasoning    extend? 

Generalize  the  truth  reached. 

Exercises. 

364.  Considering  only  the  relation  of  space  enclosed  to 
amount  of  wall,  what  would  be  the  most  economical  form  for 
the  ground  plan  of  a  house  ? 

365.  Why  is  the  most  economical  form  for  piping  that 
with  a  circular  cross-section  ? 

366.  Of. all  As  in  a  given  circle,  what  is  the  shape  of  the 
one  of  maximum  area?     Prove  your  work. 

367.  A  cross-section  of  a  bee's  cell  is  a  regular  hexagon. 
Show  that  this  is  the  best  form  for  securing  the  greatest 
capacity  with  a  given  amount  of  wax  (perimeter). 


228  PLANE  GEOMETRY,  BOOK  V. 

368.  Find  a  point  in  a  given  straight  line  such  that  the 
tangents  drawn  from  it  to  a  given  circle  contain  the  maxi- 
mum angle. 

369.  A  straight  ruler,  1  foot  long,  slips  between  the  2 
edges  of  a  rectangle.  Find  the  position  of  the  ruler  when  the 
A  thus  formed  is  a  maximum.     What  is  its  area  ? 

370.  Of  all  As  of  a  given  base  and  area,  show  which  A 
has  the  greatest  vertical  Z  . 

371.  Of  3  similar  figures  constructed  on  the  3  sides  of  a 
rt.  A>  the  figure  constructed  on  the  hypotenuse  is  equivalent 
to  the  sum  of  the  other  2  figures. 

372.  Of  all  parallelograms  of  a  given  base  and  area, 
which  has  the  least  perimeter  ?     Prove. 

373.  Given  a  square  and  a  rectangle  of  the  same  area. 
Compare  their  perimeters. 

374.  What  is  the  largest  rectangle  whose  dimensions 
are  the  two  segments  of  a  line? 

375.  From  a  given  point  without  a  circle,  draw  a  secant 
whose  outer  segment  is  a  minimum.  What  about  the  inner 
segment? 

376.  Side  of  a  regular  hexagon  :=^  1;  it  is  required  to 
find  the  sides  of  a  rectangle  that  shall  exactly  enclose  it,  and 
to  find  the  area  of  the  hexagon  and  the  area  of  the  rectangle, 
and  the  ratio  between  them. 


SELECTED  EXAMINATION   PAPERS.  229 


SELECTED    EXAMINATION    PAPERS    IN    PLANE 

GEOMETRY  SET  FOR  ADMISSION  TO  SOME 

OF    THE    LEADING    COLLEGES    OF 

THE    UNITED   STATES. 


Massachusetts  Institute  of*TechnoIogfy,  September,  J  898. 
Every  reason  must  be  stated  in  full. 

1.  If  two  Straight  lines  are  cut  by  a  third  so  as  to  make 
the  alternate  interior  angles  equal,  the  two  lines  are  parallel. 

2.  If  two  triangles  have  two  sides  of  one  equal  respect- 
ively to  two  sides  of  the  other,  but  the  included  angle  ot  the 
first  greater  than  the  included  angle  of  the  second,  the  third 
side  of  the  first  is  greater  than  the  third  side  of  the  second. 

3.  The  diagonals  of  a  rhombus  bisect  each  other  at  right 
angles. 

4.  In  the  same  circle,  or  equal  circles,  equal  chords  are 
equally  distant  from  the  center. 

5.  The  angle  between  a  secant  and  a  tangent  is  meas- 
ured by  one-half  the  difference  of  the  intercepted  arcs. 

6.  Any  two  rectangles  are  to  each  other  as  the  products 
of  their  bases  by  their  altitudes. 

7.  The  area  of  a  circle  is  equal  to  one-half  the  product 
of  its  circumference  and  radius. 

8.  A  regular  hexagon,  A  B  C  D  E  F,  is  inscribed  in  a  circle 
whose  radius  is  4.     Find  the  length  of  the  diagonal  A  C. 


Cornell,  1898. 

1.  Through  a  given  point,  P,  without  the  line,  one  and 
only  one  perpendicular  can  be  drawn  to  a  given  straight  line, 
AB. 

2.  Show  how  to  bisect  a  given  angle. 

3.  Given  a  triangle;  find  the  center  of  the  circumscribed 


230  PLANE  GEOMETRY,  BOOK  V. 

circle,   the   center   of  the    inscribed  circle,   and  the  median 
center. 

4.  Construct  a  pentagon  similar  to  a  given  pentagon 
when  the  sum  of  the  sides  is  given. 

5.  If  a  square  and  a  rhombus  have  equal  perimeters,  and 
the  altitude  of  the  rhombus  is  four-fifths  its  side,  compare  the 
areas  of  the  two  figures. 

6.  Find  the  area  of  a  trapezoid  of  which  the  bases  are  a 
and  b,  and  the  other  sides  are  equal  to  c. 

7.  Compare  the  areas  of  an  inscribed  and  circumscribed 
hexagon  about  a  given  circle. 


Harvard,  June,  1896. 

One  question  may  be  omitted. 
[In  solving  problems,  use  for  tt  the  approximate  value  3f.] 

1.  Prove  that  if  two  oblique  lines  drawn  from  a  point  to 
a  straight  line  meet  this  line  at  unequal  distances  from  the 
foot  of  the  perpendicular  dropped  upon  it  from  the  given 
point,  the  more  remote  is  the  longer. 

2.  Prove  that  the  distances  of  the  point  of  intersection 
of  any  two  tangents  to  a  circle  from  their  points  of  contact  are 
equal. 

A  straight  line  drawn  through  the  center  of  a  certain 
circle  and  through  an  external  point,  P,  cuts  the  circumference 
at  points  distant  8  and  18  inches  respectively  from  P.  What 
is  the  length  of  tangent  drawn  from  P  to  the  circumference  ? 

3.  Given  an  arc  of  a  circle,  the  chord  subtended  by  the 
arc,  and  the  tangent  to  the  arc  at  one  extremity,  show  that  the 
perpendiculars  dropped  from  the  middle  point  of  the  arc  on 
the  tangent  and  chord,  respectively,  are  equal. 

One  extremity  of  the  base  of  a  triangle  is  given  and  the 
center  of  the  circumscribed  circle.  What  is  the  locus  of  the 
middle  point  of  the  base  ? 


SELECTED  EXAMINATION    PAPERS.  231 

4.  Prove  that  in  any  triangle  the  vSquare  of  the  side 
opposite  an.  acute  angle  is  equal  to  the  sum  of  the  squares  of 
the  other  two  sides  diminished  by  twice  the  product  of  one  of 
those  sides  and  the  projection  of  the  other  upon  that  side. 

Show  very  briefly  how  to  construct  a  triangle  having 
given  the  base,  the  projections  of  the  other  sides  on  the  base, 
and  the  projection  of  the  base  on  one  bf  ihese  sides. 

5.  Show  that  the  areas  of  similar  triangles  are  to  one 
another  as  the  areas  of  their  inscribed  circles. 

The  area  of  a  certain  triangle  the  altitude  of  which  is  ^'^, 
is  bisected  by  a  line  drawn  parallel  to  the  base.  What  is  the 
distance  of  this  line  from  the  vertex? 

6.  Two  flower-beds  have  equal  perimeters.  One  of  the 
beds  is  circular  and  the  other  has  the  form  of  a  regular  hexa- 
gon. The  circular  bed  is  closely  surrounded  by  a  walk  7  feet 
wide  bounded  by  a  circumference  concentric  with  the  bed. 
The  area  of  the  walk  is  to  that  of  the  bed  as  7  to  9.  Find 
the  diameter  of  the  circular  bed  and  the  area  of  the  hexagonal 
bed. 


Yale,  June,  J  896. 
GEOMETRY  (A). 

TIME,    ONE    HOUR. 


1.  The  sum  of  the  three  angles  of  a  .triangle  is  equal  to 
two  right  angles. 

2.  Construct  a  circle  having  its  center  in  a  given  line 
and  passing  through  two  given  points. 

3.  The  bisector  of  the  angle  of  a  triangle  divides  the  op- 
posite side  into  segments  which  are  proportioned  to  the  two 
other  sides. 

4.  If  two  angles  of  a  quadrilateral  are  bisected  by  one 
of  its  diagonals,  the   quadrilateral  is  divided  into  two  equal 


232  PLANE  GEOMETRY,  BOOK  V. 

triangles  and  the  two  diagonals  of  the  quadrilateral  are  per- 
pendicular to  each  other. 

5.     The  circumferences  of  two  circles  are  to  each  other 
as  their  radii      (Use  the  method  of  limits.) 


Yale,  June,  \Z96. 
GEOMETRY  (B). 

TIME   ALI.OWED,    FORTY-FIVE   MINUTES. 

1.  A  tree  casts  a  shadow  90  feet  long,  when  a  vertical 
rod  6  feet  high  casts  a  shadow  4  feet  long.  How  high  is  the 
tree? 

2.  The  distance  from  the  center  of  a  circle  to  a  chord  10 
inches  long  is  12  inches.  Find  the  distance  from  the  center 
to  a  chord  24  inches  long. 

3.  The  diameter  of  a  circular  grass  plot  is  28  feet.  Find 
the  diameter  of  a  grass  plot  just  twice  as  large.  (Use  loga- 
rithms.) 

4.  Find  the  area  of  a  triangle  whose  sides  are^  =  12.342 
metres  b  =  31.456  metres  <:=  24.756  metres,  using  the  formula 

Area  =  sj  s{s-a)  {s-b)  (s~c)  where  s  =  ^^ — -.      (Use  loga- 
rithms.) 


Princeton,  June,  t896. 

State  what  text-book  you  have  read  and  how  much  of  it. 

1.  Prove  that  the  sum  of  the  three  angles  of  a  triangle  is 
equal  to  two  right  angles;  and  that  the  sum  of  all  the  interior 
angles  of  a  polygon  o(  n  sides  is  equal  to  (n — 2)  times  two 
right  angles. 

2.  Show  that  the  portions  of  any  straight  line  intercepted 
between  the  circumferences  of  two  concentric  circles  are  equal. 

3.  Define  similar  polygons  and  show  that  two  triangles 
whose  sides  are  respectively  parallel  or  perpendicular  are 
similar  polygons  according  to  the  definition. 


SELECTED  EXAMINATION   PAPERS.  233 

4.  Prove  that,  if  from  a  point  without  a  circle  a  secant 
and  a  tangent  are  drawn,  the  tangent  is  a  mean  proportional 
between  the  whole  sejant  and  its  external  segment. 

5.  Prove  what  the  area  of  a  triangle  is  equal  to;  also  of 
a  trapezoid;  also  of  a  regular  polygon.  Define  each  of  the 
figures  named. 

6.  Explain  how  to  construct  a  triangle  equivalent  to  a 
given  polygon. 

7.  Prove  that  of  all  isoperimetric  polygons  of  the  same 
number  of  sides,  the  maximum  is  equilateral. 


Johns  Hopkins  University,  October,  J896. 

1.  Prove  that  the  bisectors  of  the  two  pairs  of  vertical 
angles  formed  by  two  intersecting  lines  are  perpendicular  to 
each  other. 

2.  Show  that  through  three  points  not  lying  in  the  same 
straight  line  one  circle,  and  only  one,  can  be    made  to  pass. 

3.  The  bases  of  a  trapezoid  are  16  feet  and  10  feet  re- 
spectively; each  leg  is  5  feet.  Find  the  area  of  the  trapezoid. 
Also  find  the  area  of  a  similar  trapezoid,  if  each  of  its  legs  is 
3  feet. 

4.  Define  regular  polygon.  Prove  that  every  equi- 
angular polygon  circumscribed  about  a  circle  is  a  regular 
polygon. 

5.  Prove  that  the  opposite  angles  of  a  quadrilateral  in- 
scribed in  a  circle  are  supplements  of  each  other. 

6.  Construct  a  square,  having  given  its  diagonal. 

7.  Prove  that  the  area  of  a  triangle  is  equal  to  half  the 
product  of  its  perimeter  by  the  radius  of  the  inscribed  circle. 

8.  What  is  the  area  of  a  ring  between  two  concen- 
tric circumferences  whose  lengths  are  10  feet  and  20  feet 
respectively? 


234  PLANE  GEOMETRY,  BOOK  V. 


Sheffield  Scientific  School,  June,  tS96* 

[NoTK. — State  at  the  head   of  your  paper  what  text-book  you  have 
studied  on  the  subject  and  to  what  extent  ] 

1.  Two  angles  whose  sides  are  parallel  each  to  each  are 
either  equal  or  supplementary.  When  will  the}^  be  equal,  and 
when  supplementary  ? 

2.  An  angle  formed  by  two  chords  intersecting  within 
the  circumference  of  a  circle  is  measured  by  one-half  the  sum 
of  the  intercepted  arcs. 

3.  A  triangle  having  a  base  8  inches  is  cut  by  a  line  par- 
allel to  the  base  and  6  inches  from  it.  If  the  base  of  the 
smaller  triangle  thus  formed  is  5  inches,  find  the  area  of  the 
larger  triangle. 

4.  Construct  a  parallelogram  equivalent  to  a  given 
square,  having  given  the  sum  of  its  base  and  altitude.  Give 
proof. 

5.  What  are  regular  polygons  A  circle  may  be  circum- 
scribed about,  and  a  circle  may  be  inscribed  in,  any  regular 
polygon. 


The  University  of  Chicago,  September,  tZ96* 

TIME   ALLOWED,    ONE    HOUR   AND   FIFTEEN    MINUTES. 

[When   required,   give  all  reasons  in  full  and  work  out  proofs  and 
problems  in  detail  ] 

1.  Show  that  if  on  a  diagonal  of  a  parallelogram  two 
points  be  taken  equally  distant  from  the  extremities,  and  these 
joints  be  joined  to  the  opposite  vertices  of  the  parallelogram, 
the  four-sided  figure  thus  formed  will  be  a  parallelogram. 

2.  State  and  prove  the  converse  of  the  following  theorem. 
In  the  same  circle,  equal  chords  are  equally  distant  from 

the  center. 

3.  Given  a  circle,  a  point,  and  two  straight  lines  meet- 
ing in  the  point  and  terminating  in  the  circumference  of  the 


SELECTED  EXAMINATION   PAPERS.  235 

circle.     State  what  four  Hues  or  segments  form  a  proportion 
and  in  what  order  they  must  be  taken: 

(1)  When  the  point  is  outside  the  circle,  and 
(a)  both  lines  are  secants, 

(d)  one  line  is  a  secant,  and  the  other  a  tangent, 
(c)  both  lines  are  tangents. 

(2)  When  the  point  is  within  the  circle,  and  the  two 
lines  are  chords. 

Prove  in  full  (1)  (a).  Show  that  (1)  (c)  is  a  limiting  case 
of  (1)  (a). 

4.  To  a  given  circle  draw  a  tangent  that  shall  be  per- 
pendicular to  a  given  line. 

5.  Show  how  to  construct  a  triangle,  having  given  the 
base,  the  angle  at  the  opposite  vertex,  and  the  median  from 
that  vertex  to  the  base.  Discuss  the  cases  depending  upon 
the  length  of  the  given  median. 


"Wcllcsley  College,  June,  iS95* 

1.  An  angle  formed  by  two  tangents  is  how  measured? 
Prove. 

2.  The  diagonals  of  a  rhombus  bisect  each  other  at  right 
angles. 

3.  (a)  If  a  line  bisects  an  angle  of  a  triangle  and  also 
bisects  the  opposite  side,  the  triangle  is  isosceles. 

(d)  State  and  demonstrate  the  general  case  for  the  ratio 
of  the  segments  of  the  side  opposite  to  a  bisected  angle. 

4.  With  a  given  line  as  a  chord,  construct  a  circle  so  that 
this  chord  shall  subtend  a  given  inscribed  angle. 

5.  (a)  On  a  circle  of  4  feet  radius,  how  long  is  an  arc  in- 
cluded between  two  radii  forming  an  angle  of  20°?  Prove, 
deriving  the  formula  employed. 

id)  Find  the  area  of  the  regular  circumscribed  hexagon 
of  a  circle  whose  radius  is  1. 

,6.  Two  similar  triangles  are  to  each  other  as  the  squares 
of  their  homologous  sides. 


23«  SOLID   GEOMETRY,  BOOK  VI. 


BOOK  VI. 
SOLID  GEOMETRY. 

Preliminary  Discussion. 
378. 

1.  (1)  Construct  three  plane  figures;  two  not  plane. 
(2)  Why  call  the  ^rst p/ane  figures?  (3)  What  is  a  plane? 
What  are  its  limits?  How  illustrate  or  represent  a  plane? 
(4)  How  test  a  plane  surface  ?  a  curved  surface  ? 

2.  What  is  your  idea  of  space  ? 

3.  Explain  the  terms  ^nile  and  infiyiite.  {Definition: 
A  finite  (see  §  1)  portion  of  space  regarded  as  separated  from 
the  rest  is  called  a  solid.)  Draw  distinction  between  a  phys- 
ical solid  and  a  geometrical  solid. 

4.  What  does  a  surface  do  to  space?  What  does  a 
closed  surface  separate?  What  separates  one  portion  of  space 
from  another  ? 

5.  Suppose  one  definite  portion  of  surface  is  separated 
from  the  rest,  what  kind  of  a  line  do  we  find  ? 

6.  What  separates  one  part  of  a  line  from  another? 

7.  What  can  you  assert  of  a  point  ?  Can  we  compare 
two  points  ?     How  may  a  point  be  determined  ? 

8.  Explain  the  difference  between  a  square  and  a  cube; 
a  circle  and  a  sphere;  a  triangle  and  a  triangular  prism. 

9.  What  is  the  difference  between  plane  figures  and  solid 
figures  ? 

10.  What  is  meant  by  the  position  of  a  plane  ?  Illustrate 
'with  cardboard. 

11.  How  many  planes  may  be  made  to  pass  through  a 


LINES  AND   PLANES  IN  SPACE.  237 

given  point?  two  points?  a  straight  line?  three  points?     Use 
pins  to  illustrate  lines  and  points. 

12.  Does  one  line  determine  a  plane?  Illustrate  with  a 
card. 

13.  Can  you  recall  your  definition  of  a  postulate  f  State 
a  postulate  about  changing  the  position  of  a  figure.  Suppose 
one  point  of  a  figure  to  be  fixed,  how  does  it  affect  the  figure  ? 
Suppose  a  second  point  fixed?  a  third? 

14.  What  is  the  locus  of  a  mo/ing  point?  Is  a  point  a 
part  of  a  line?  (How  many  points  make  aline  one  foot  long?) 
What  determines  a  straight  line  ? 

15.  How  many  points  determine  a  curved  line?  How 
many  straight  lines  are  determined  by  two  points  ? 

16.  Think  of  two  points  on  a7iy  surface.  How  many 
lines  may  be  passed  through  these  two  points?  Can  there  be 
between  two  points  on  any  surface  one  line  passing  along  the 
surface  shorter  than  all  the  others?  Such  a  line  is  called  a 
direct  or  geodesic  line. 

17.  What  must  we  have  in  order  to  locate  one  point  on 
a  surface  relative  to  another  ?  (A  surface  is  a  magnitude  of 
two  dimensions.) 

18.  How  many  items  must  be  considered  to  locate  two 
points  relatively  in  space?  Illustrate.  Define  a  plane  sur- 
face; a  curved  surface. 

379. 

1.  When  is  a  straight  line  ||  to  a  plane? 

2.  When  are  planes  parallel  ? 

380. 
Proposition  I. 

(I)  Can  you  think  of  an  easy  way  to  pass  a  plane  through 
a  line  and  a  point  ?  Illustrate.  How  many  planes  may  be 
passed  through  a  given  line  and  a  given  point?  Can  you  show 
that  this  plane  is  fixed? 


238  SOLID   GEOMETRY,   BOOK  VI. 

(2)  Think  of  three  points,  m,  71,  o,  not  in  the  same 
straight  line.  What  is  one  way  to  pass  a  plane  through  these 
three  points  ?  How  many  planes  may  be  passed  through 
these  three  points? 

(3)  Draw  two  \\  lines,  A  B  and  C  D.  Pass  a  plane,  m  n, 
through  A  B.  Can  you  pass  the  same  plane  through  CD? 
Will  these  two  lines  fix  the  position  of  ;;2  « ?  Why?  Illus- 
trate with  straight  wires  and  a  card. 

(4)  Draw  two  lines,  A  B  and  C  D,  that  intersect. 
Pass  a  plane  through  A  B.  Can  you  pass  the  same  plane 
through  CD?  Is  the  plane  fixed?  How  many  planes  can 
you  pass  through  these  two  lines  ? 

Make  a  summary  of  the  conditions  which  will 
locate  a  plane. 

Call  it  Prop.  I. 

Preliminary  Discussion. 

381. 

The  beginner  in  Solid  Geometry  will  find  his  ideas  grow- 
ing clearer  if  he  will  use  pieces  of  card  board  to  represent 
planes  and  straight  wires  to  represent  lines  and  points.  To 
illustrate:  Take  two  postal  cards  and  cut  each  half  in  two, 
and  then  fit  the  cards  together.  This  will  represent  to  the 
mind  the  intersection  of  two  planes  which  appears  to  be  a 
straight  line.  A  darning-needle  or  hat-pin  put  through  these 
cards  will  represent  a  line  piercing  two  planes.  An  ordinary 
room  will  illustrate  some  of  the  problems  in  Solid  Geometry. 
The  pupil  will  understand  that  these  are  simply  aids  to  the 
imagination.  When  the  learner  is  able  to  see  planes  inter- 
secting, lines  making  angles,  solids  cut  by  planes,  solids  cut 
by  other  solids,  without  the  aid  of  physical  illustrations,  he  is 
making  a  good  start  in  Solid  Geometry. 


LINES  AND   PLANES  IN  SPACE.  239 

Exercise. 

377.  Name  some  problems  in  Solid  Geometry  which  the 
carpenter  must  solve;  some  the  plumber  must  solve;  some  the 
brickmason  must  solve. 

382. 

In  how  many  points  may  a  straight  line  intersect  a  plane  ? 
Prove  ? 

Sug.  If  we  suppose  the  line  to  meet  the  plane  in  two 
points,  what  definition  is  violated  ? 

383. 

Proposition  II. 

Suppose  two  planes  meet,  what  can  you  prove 
of  their  intersection  ? 

Sug.  Take  any  two  points  in  the  line  of  intersection 
and  join  them.  What  is  the  line  ?  Where  does  it  lie  ?  Il- 
lustrate using  a  pair  of  scissors  and  two  pieces  of  cardboard. 
Draw   figures  also.     Write   a  neat  proof. 

Generalize  these  truths,  calling  the  statement  Prop.  II. 

384. 

Proposition   III. 


In  the  figure  let  j  be  a  line  ||  the  line  x  in  the  plane  M  N. 
Required  to  find  how  y  is   related  to   the  plane 


M  N, 


240  SOLID   GEOMETRY,   BOOK   VI. 

Sug.     Pass  a  plane  through  x  and  y. 

Suppose y  could  meet  M  N  (in  P,  say).     In  what  plane 
would   it  meet  M  N?  in  what  line?     Any  violation  of  con- 
ditions?    Write  a  careful  proof.     Generalize. 
Call  this  Prop.  III. 

385. 
Proposition  IV.- 

In  the  figure  above  let  j^^  be  any  line  ||  to  M  N.     Pass  any 
plane  through  y  intersecting  M  N. 

Write  a  full  proof  showing  how  the  intersection 
of  the  two  planes  is  related  to  y. 
Call  the  generalization  Prop.  IV. 

386. 

Cor.     Suppose  that  a  line,  <:,  is  ||  toy,  in  §  385,  and  passes 
through  a  point,  P,  in  the  plane  M  N. 

Prove  where  the  line  c  lies. 

Sug.     Can   you  prove   it   coincident   wither?     Write    a 
statement  of  this  corollary. 

387. 

Proposition  V. 

Can  you  show  that  if  two  lines  are  ||  to  a  third 
line  they  are  ||  to  each  other 
Use  cards  to  illustrate. 


In  the  figure  let  x  and  y  be  ||  z. 

Call  some  point  on  y,  P.  Pass  planes  through  x  z  ',y  z  ;  x 
and  P.     Call  the  line  intersecting  y  z  and  x  P,  //.     Can  you 


LINES  AND   PLANES  IN  SPACE.  241 

show  that  ij  coincides  with  y  ?    Write  your  proof  and  state 
theorem. 

Call  it  Prop.  V. 

Can  you  prove  this  theorem  by  projecting  one  line  ? 

Z  II  plane  ;r  R     [  ?  ] 

Z||>'[?] 

y  lies  in  plane  xV.     [  ?  ] 

.;»:  11  line  J  P  L     [?] 

y  lies  in  plane  x  P  and  passes  through  P; 

.  •  .  J  P  I  2iX\Ay  must  coincide.     [  ?  ]     Q.  E.  D. 


Exercises. 

378.  Can  you  show  that  any  theorem  in  Plane  Geometry 
in  regard  to  a  A  is  true  also  in  Solid  Geometry  ? 

Illustrate  your  exercises  with  good  drawings. 

379.  If  a  plane  is  passed  through  each  of  2  ||  lines,  the 
intersection  of  the  planes  is  ||  to  each  of  the  lines. 

380.  If  8  planes  meet  in  3  lines,  the  intersections  either 
meet  in  a  point  or  they  are  ||. 

381.  If  each  of  the  two  intersecting  lines  is  ||  to  a  plane, 
the  plane  of  these  lines  is  ||  to  the  first  plane. 

382.  Parallel  lines  included  between  ||  planes  are  equal. 

383.  Construct   through   a  given   point  a  plane  ||  to  a 
given  plane. 

384.  Construct  through  a  given  line  a  plane  ||  to  a  given 
line. 

385.  Construct  through  a  given  point  a  plane  H  to  each 
2  lines.     Is  this  ever  impossible  ?     Discuss. 


242  SOLID   GEOMETRIC  BOOK  VI. 

388. 
Proposition  VI. 


Stand  a  book  on  the  desk  partly  open  and  suppose  the 
pages  to  represent  parallelograms.  Compare  the  angle  made 
by  the  two  top  edges  with  the  angle  made  by  the  two  cor- 
responding lower  edges.  Can  you  illustrate  the  same  truth 
in  the  room  ? 

In  the  figure  suppose  a  and  b  two  intersecting  lines  and 
a'  and  b'  two  intersecting  lines  respectively  \\\.o  a  and  b. 

Compare  their  Z  s. 

Call  the  intersections  O  and  O'.  Join  these  points.  From 
any  points  on  a  and  by  as  P  and  Q,  erect  ||s  to  O  O'. 

(1)  Can  you  pass  a  plane  through  «,  a\  and  O  O'? 

(2)  Will  the  parallel  from  P  meet  a'  ?  Why  ?  L^etter  the 
point,  P'. 

(3)  Will  parallel  through  Q  intersect  b'>  I^etter  inter- 
section Q  ? 

(4)  Compare  O  P  and  O'  P';  O  Q  and  O'  Q'. 

(5)  Join  P  Q  and  P'  Q'.    How  are  P  Q  and  P'  Q'  related? 

(6)  Compare  Z  P  O  Q  with  Z  F  O'  Q'. 

(7)  Show  that  Z  P  O  Q  is  supplementary  to  tvo  angles 
formed  by  a'  and  b'. 

Draw  conclusion,  and  call  it  Prop.  VI. 


LINES  AND   PLANES  IN  SPACE.  243 

389. 

Prove  that  if  each  of  two  iutersecting  lines  is  || 
to  a  plane,  the  plane  of  these  lines  is  ||  to  the  first 
plane. 

Use  reductio  ad  absurdum.  Suppose  the  plane  of  the  in- 
tersecting lines  meets  the  given  plane  in  line  x.  Then  each 
of  the  intersecting  lines  meets  the  plane.     (Pupil  finish.) 

Call  this  Prop.  VII. 

Exercises. 

386.  If  a  line  cuts  one  of  two  ||  lines,  must  it  cut  the 
other?     Are  the  corresponding  Z  s  equal. 

387.  Prove  that  ||  lines  included  between  ||  planes  are 
equal. 

388.  If  2  II  lines  intersect  a  plane,  compare  the  angles 
formed. 

389.  If  a  straight  line  intersects  2  ||  planes,  compare  the 
angles  formed. 

390.  If  a  line  is  ||  to  each  of  2  intersecting  planes,  how 
is  it  related  to  their  intersection? 

Definitions. 

390. 

The  distance  from  a  point  to  a  plane  is  the  perpendicular 
distance. 

391. 

The  point  where  a  1  meets  a  plane  is  called  the  foot  of  the 
perpendicular, 

392. 

A  line  is  said  to  be  1,  or  normal  to  a  plane,  when  it  is  ±  to 
every  line  in  that  plane  which  passes  through  its  foot.  When 
is  a  line  oblique  to  a  plane? 


244 


SOLID   GEOMETRY,   BOOK  VI. 


393. 

Lines  or  points  which  lie  in  the  same  plane  are  coplanar 

394. 

(1)  Three  or  more  points  which  lie  in  the  same  line  are 
said  to  collhiear. 

(2)  A  line  is  |1  to  a  plane  if  it  cannot  meet  it  however  far 
produced.     The  plane  is  said  to  be  1 1  to  the  line. 

395. 

The  piojection  of  a  point  on  a  plane  is  the  foot  of  the  1 
from  the  point  to  the  plane.  Illustrate  with  your  pencil  and 
desk. 

396. 

The  projection  of  a  line  on  a  plane  is  a  straight  line  join- 
ing the  projections  of  the  extremities  of  the  line  on  the  plane. 
In  the  figure,  a,  b,  c,  d  are  projections  of  the  points  A,  B,  C,  D 
and  line  a  d  \v\  the  projection  of  line  A  D  on  the  plane  M  N. 


397. 

The  smaller  angle  formed  by  a  line  and  its  projection  is 
called  the  inclination  of  the  line  to  the  plane. 

398. 

The  angle  which  a  line  makes  with  a  plane  is  the  angle 
which  it  makes  with  its  projection.  Illustrate  with  pencil  and 
desk. 


LINES  AND  PLANES  IN  SPACE. 


245 


399. 

The  plane  which  a  line  makes  with  its  projection  is  called 
the  projecting  plane. 

400. 

Where  are  all  the  common  points  of  2  planes? 

EXERCISKS. 

391.  How  many  planes  are  determined  by  6  points,  3 
being  collinear? 

392.  How  many  planes  in  general  are  determined  by  4 
points  in  space,  no  3  being  collinear? 

393.  A  point,    P,   is  in    three  planes,  P,    Q,    R.     Is  it 
necessarily  fixed? 

401. 

Proposition  VII. 


In  the  figure  suppose  line  ^  1  to  both  b  and  d. 
How  is  a  related  to  auy  other  line,  as  c^  lying  in  the 
plane  of  b  and  d  and  passing  through  their  inter- 
section ? 

Sug.  I.  On  line  a  make  O  P  =  O  P';  draw  any  line  cut- 
ting b,  c,  of  in  E,  F,  G,  and  join  points  with  P  and  P'. 


246  SOLID    GEOMETRY,   BOOK  VI. 

Sug.  11.     Compare  (1)   /\s    E  G  P'  and  E  G  P;  (2)  As 
E  F  P  and  E  F  P';  (3)  and  As  P  O  F  and  P'  O  F. 
How  then  is  c  related  to  a  ? 
State  the  general  truth,  Prop.  VII. 

402. 

Cor.  I.  If  a  line  is  i  to  each  of  2  intersecting 
lines,  show  from  the  figure  above  how  it  is  related 
to  their  plane. 

403. 

Cor.  II.  Prove  what  line  of  all  those  drawn  from 
a  given  point  to  a  plane  is  the  shortest.  Which  lines 
are  equal? 

404. 

Cor.  HI,  What  is  the  locus  of  points  equally 
distant  from  2  points? 

405. 

Cor.  IV.  What  is  the  locus  of  straight  lines 
which  cut  a  given  straight  line  at  a  given  point  at 
right  angles? 

Note. — The  proof  that  «  is  1  to  every  straight  line  that 
meets  it  in  the  plane  R  S  is  due  the  great  French  mathemati- 
cian lyCgendre.     (See  his  biography.) 

Exercises. 

394.  Through  a  given  point  in  a  plane,  how  many  j^s 
may  be  erected  to  the  plane  ? 

395.  Through  a  given  point  in  a  line,  how  many  ±  planes 
can  be  drawn  to  that  line  ? 

396.  Suppose  the  hand  of  a  clock  to  be  I  to  its  moving 
axle.    Show  what  kind  of  a  figure  it  describes  in  revolving. 


LINES  AND   PLANES  IN  SPACE. 


247 


397.  How  many  planes  are  determined  by  5  concurrent 
lines,  no  3  of  which  are  coplanar  ?     By  x  lines  ? 

398.  If  a  line  cuts  one  of  2  H  lines,  must  it  cut  the  other? 
If  it  does,  are  the  corresponding  Zs  equal? 


406. 
Proposition  VIII. — Theorem. 

B 


In  the  figure  suppose  B  P  i  to  the  three  lines 
P  H,  P  I,  P  J  at  their  point  of  concurrence.  Can 
you  show  how  these  three  lines  are  situated? 

Sug.  1.  Let  R  S  be  the  plane  determined  by  P  H  and 
P  I,  and  M  Q  the  plane  determined  by  B  P  and  P  J. 

Sug.  2.  Suppose  P  J  is  not  in  R  S  and  let  P  N'  be  the 
intersection  of  the  two  planes.  What  does  this  supposition 
involve?  What  can  you  now  state  concerning  the  three  lines 
which  are  1  to  B  P  at  their  point  of  concurrence? 

Write  the  general  truth  and  call  it  Prop.  VIII. 

407. 

Cor.  /.  Show  that  lines  1  to  the  same  line  at 
the  same  point  are  coplanar. 


248 


SOLID   GEOMETRY,   BOOK  VI. 


408. 

Cor.  II.  Through  a  given  point  in  a  plane  how 
many  Is  can  be  erected  ? 

l^Hint. — Suppose  two  perpendicular  lines  can  be  erected. 
Pass  a  plane  through  those  two  lines.     What  follows?] 

409. 

Cor.  III.  Can  you  show  from  the  figure  above 
that  only  one  plane  can  be  passed  through  a  given 
point  in  a  line  1  to  that  line? 

410. 

Proposition  IX. 


Given:     The  line  P  Q  1   to  plane  R  U  and  P'  Q'  ||  P  Q- 
How  is  P' Q' related  to  R  U? 

Sug.  Draw  P  S,  P  T  in  the  plane  through  P,  and 
through  P'draw  P'  S',  F  T'  ||  P  S  and  P  T  respectively. 

Compare  Z  Q'  P'  S'  with  /  Q  P  S,  Z  Q'  P'T'  with 
Q  P  T.     How  then  is  Q'  P'  related  to  the  plane  R  U  ? 

Write  the  general  truth,  and  call  it  Prop.  IX. 


LINES  AND  PLANES  IN  SPACE. 


249 


411. 
Proposition  X. 


Suppose  A  B  and  C  D  1  to  plane  M  N.  how  are  these 
two  lines  related  ? 

Sug.  Suppose,  if  possible,  a  third  line,  D  E,  to  be  erected 
II  A  B. 

What  do  you  know  of  E  D  ?  Is  there  any  previous 
proposition  violated  ? 

Suppose  3  lines  1  to  the  same  plane.  Show  how  the  lines 
are  related  ?     Compare  with  Prop.  V. 

412. 
Cor.     How  many  is  from  a  point  to  a  plane? 

413. 

Proposition  XI. 


Given:    The  points  A  and  B   in    the  line  x  y,   which 
pierces  plane  M  N.     Project  these  points  in  the  plane  M  N. 


250  SOLID   GEOMETRY,  BOOK  VI. 

Join  the  projections  of  A  and  B.  Call  the  line  A'  B'.  Does 
the  projection  of  any  other  point  in  the  line  xy,  C  lie  in  the 
same  line  with  A'B7 

Sug.  What  can  you  say  of  A  A?  B  B?  C  C?  Pass  a 
plane  through  B  B',  A  A'.  Is  the  point  C  in  this  plane?  C  C? 
Does  A'  Q!  B'  lie  in  the  straight  line  A'  B7 

Does  the  line  determined  by  the  projections  of 
any  two  points  of  a  given  line  give  its  projection  ? 
Generalize. 

This  will  be  Prop.  XI. 

Can  you  prove  this  directly  from  t  he  definition  of  the 
projection  of  a  line? 

414. 

Cor,  Suppose  a  line  20  dm.  long  pierces  a  plane- 
Does  it  meet  its  projection?  Where?  Compare  tbe 
lengths  of  tbe  projections  if  tbe  line  were  (1)  pro- 
jected on  tbe  plane  pierced;  (2)  projected  on  a  plane 
o  wbicb  it  i  s  parallel. 

Exercises. 

399.  Are  lines  which  make  equal  angles  with  a  given 
line  always  ||  ?     Illustrate  your  answer. 

400.  A  line  equals  its  projection;  construct  a  figure  to 
compare  the  positions  of  the  two  lines. 

401.  Can  you  show  when  the  projection  of  a  line  is  half 
the  length  of  the  line?     When  is  it  zero? 

402.  Two  II  Hues  have  their  projections  in  the  vSame 
plane.     What  can  you  show  of  their  projections  ? 

403.  If  the  projections  of  two  lines  are  1|  or  coincident, 
prove  whether  or  not  the  lines  are  1|. 

404.  If  the  projection  of  a  line  24  inches  long  is  10 
inche?^  find  the  projection  on  the  same  plane  of  a  parallel 
line  32  inches  long. 


LINES  AND   PLANES  IN  SPACE. 

415. 

Proposition  XXL 


251 


In  the  figure  let  A  C,  D  F  be  any   two  lines  cut  by  three 
II  planes,  R  S,  P  Q,  M  N,  in  the  points  A,  B,  C,  D,  E,  F. 

Compare  the  segments  of  the  lines. 

S^^^-  Join  A,  F.  Pass  a  plane  through  A  C  and  A  F. 
and  F  A  and  F  D.  Why  can  we  do  this  ?)  Let  C  F  and  B  G 
be  the  intersections  made  by  the  first  plane  with  the  others 
and  A  D  and  G  E  the  intersections    made   by   the  second 

A  B        D  E  . 


B  C 


E  F 


plane.      Can  you  now  show  that 
Write  Prop.  XII. 

416. 


Cor.  I.  Suppose  two  lines  are  cut  by  any  number  ol  || 
planes.  What  can  you  say  or  prove  of  the  corresponding 
segments? 


262  SOLID   GEOMETRY,  BOOK  VI. 

417. 

Cor.  II.     If  X   straight   lines  be   cut  by   three  ||  planes, 
prove  how  the  corresponding  segments  are  related. 

418. 

Proposition  XIII. 


Given:  P,  a  point  without  the  plane  M  N,  and  P  O,  a 
perpendicular  to  it. 

1.  Compare  PO  with  any  other  line  drawn  from 
P  to  the  plane. 

In  how  many  ways  can  you  make  the  comparison? 
Generalize. 

2.  Given:  P  O  1  M  N  and  Z  B  =  Z  A  (angles 
of  inclination),  to  compare  P  B  and  P  A.  State  the 
converse  and  prove  it. 

Generalize. 


LINES  AND   PLANES   IN  SPACE. 


263 


3.  If  the  projections  of  two  lines  from  the  same 
point  to  the  same  plane  are  equal,  how  are  the  lines 
related  ?     State  and  prove  the  converse. 

Generalize  each. 

4.  Given  the  unequal  oblique  Zs  P  B  O  > 
P  C  O,  to  compare  P  B  and  P  C.  State  and  prove 
the  converse. 

Generalize  each. 

5.  vSuppose  the  projections  of  two  oblique  lines 
drawn  from  a  point,  P,  to  the  plane  M  N  to  be  un- 
equal, compare  the  lines  projected. 

Write  a  general  statement  comprehending  the  five  state- 
ments above.     Begin  in  this  manner:    Of  all  lines  that  can  be 

drawn  from  a  point  to  a  plane  1. ,  2. ,  8. ,  4. , 

5. . 

Exercises. 

4U5.  Parallel  line  segments  are  proportional  to  their 
projections  on  a  plane. 

40(5.  Can  you  project  two  lines  on  three  different  planes 
at  the  same  time. 

419. 
Proposition  XIV. 


Let  W  P  be  any  line  intersecting  the  plane  R  S  at  point 
P,  and  let  O  P  be  its  projection  on  R  S.  Let  Q  T  be  a  line  in 
R  S  1  O  P. 


254  SOLID   GEOMETRY,  BOOK  VI. 

Sbow  how  Q  T  is  related  to  the  given  line  W  P. 

Sug.     Measure  off  on  Q  T,  P  H  =  P  I.      Join  O  and  H, 

0  and  I,  W  and  H,  W  and  I.     Compare  As  P  O  I  and  P  O  H, 

1  W  and  H  W;  also  As  I  P  W  and  H  P  W.     How  is  W  P 
related  to  Q  T?     Generalize  the  truth  reached? 

Exercises. 

407.  If  a  line  is  ||  to  each  of  two  intersecting  planes,  how 
is  it  related  to  their  intersection?     Prove. 

408  Can  you  construct  a  plane  containing  a  given  line 
and  II  to  another  line? 

409.  If  two  II  lines  intersect  the  same  plane,  show  that 
they  are  equally  inclined  to  it. 

DiEDRAL  Angles. 

420. 

Definition:  When  any  number  of  planes  pass  through 
the  same  line,  they  are  said  to  form  a  pencil  of  planes,  and  any 
two  of  the  planes  form  a  diedral  angle. 


421. 

The  planes  are  the  faces  of  the  diedral  angle,  and  their 
intersection  the  edge. 

422. 

A  diedral  angle  may  be  designated  by  two  letters  on  its 
edge,  but  if  several  diedral  angles  have  a  common  edge,  then 
four  letters  are  necessary,  one  in  each  face  ^nd  two  on  the 
edge,  thus :  S  D  C  P,  N  D  C  M. 


LINES  AND   PLANES  IN  SPACE. 

423. 


255 


Definition:  The  plane  angle  of  a  diedral  Z  is  the  angle 
formed  by  two  straight  lines,  one  in  each  plane,  drawn  per- 
pendicular to  the  edge  at  any  given  point. 


Thus  if  B  A  and  C  A  in  the  faces  D  F  and  E  G,  re- 
spectively, are  each  IDE,  they  form  the  plane  Z  of  the 
diedral  D  E. 

424. 

By  using  your  cardboard  and  by  drawings,  illustrate  ver- 
tical diedral  angles,  adjacent  diedral  angles,  right  diedral 
angles.     Write  a  definition  of  each. 

\_Note. — The  faces  of  the  diedral  angle  are  indefinite  in 
extent,  but  for  convenience  in  study  we  take  a  limited  portion 
of  the  bounding  planes.  The  pupil  should  take  pains  in 
learning  to  draw  the  figures  in  Solid  Geometry.] 

425. 

What  plane  angle  will  be  formed  if  a  plane  be  passed 
perpendicularly  to  the  edge  of  a  diedral  angle  and  intersecting 
its  sides? 

Through  a  given  line  in  a  plane,  how  many  perpendicu- 
lar planes  can  be  passed  to  the  given  plane  ?     Why  ? 


256 


oOLID   GEOMETRY,   BOOK  VI. 


Exercises  and  Questions. 

410.  How  many  diedral  Z  s  are  formed  by  two  inter- 
secting planes. 

Can  you  show  by  illustration  how  diedral  Z  s  may  vary  ? 

411.  How  many  diedral  Zs  has  a  cube?  a  square  pyra- 
mid?  a  triangular  pyramid? 

412.  Make  a  drawing  showing  two  planes  meeting. 
What  is  the  sum  of  the  diedral  angles  formed? 

413.  Represent  two  complementary  diedral  angles. 

414.  Draw  two  parallel  planes  cut  by  an  oblique  plane. 
Name  the  equal  angles  and  pairs  whose  sums  equal  two  right 
diedral  angles.     Compare  with  §  68. 

415.  State  for  diedral  angles  what  §  73  does  for  plane 
angles. 

410.  Draw  two  diedral  angles  whose  corresponding  sides 
are  parallel.  In  how  many  ways  can  you  draw  this?  Com- 
pare the  diedral  angles 

417.  Draw  two  diedral  angles  whose  corresponding  faces 
are  perpendicular  to  each  other.  Discuss  the  diedral  angles 
formed. 

426. 
Proposition  XV. 


Suppose  R  S  1  P  Q  and  line 
tion  of  the  two  planes. 

How  is  M  N  related  to  P  Q? 


M   N  1  S  T  the  intersec- 


LINES  AND   PLANES  IN  SPACE.  257 

Sug.  At  N  draw  NO  1  S  T  in  plane  P  Q.  What  is 
the  angle  M  N  O?  Why?  What  two  lines  determine  PQ? 
Write  Prop.  XV. 

427. 

Cor,  I.  From  figure  above  show  that  a  i  to 
either  plane  at  any  point  of  S  T  lies  in  the  other 
plane. 

Exercise. 

418.  A  line  is  1  to  a  plane.  Can  you  show  that  every 
plane  passed  through  the  line  is  -L  to  the  plane. 

428. 

Cor,  II,  Suppose  two  planes  1  to  each  other. 
Can  you  show  that  a  1  from  any  point  of  the  one 
plane  to  the  other  must  lie  in  the  first  plane  ? 

429. 

Cor.  III.  Through  a  point  without  a  line  prove 
how  many  1  planes  can  be  passed  1  to  the  line. 

430. 

Proposition  XVI. 
Can  you  show  that  vertical  diedral  Z  s  are  equal  .^ 

431. 

Proposition  XVII. 
If  a  plane  is  i  the  edge  of  a  diedral  Z ,  how  is  it 
related  to  each  face  of  the  diedral  Z  ? 

17 


258 


SOLID   GEOMETRY,  BOOK  VI. 

432. 

Proposition  XVIII. 


Given:     A  B,  any  line  not  1  to  the  plane  M  N. 

How  many  planes  can  be  passed  through  A  B  _[_ 
M  N? 

{Hint. — Project  the  point  A  to  the  plane  M  N.  Pass  a 
plane  through  A  B  and  A  P.] 

Draw  conclusion  and  call  this  Prop.  XVIII. 

Exercise. 
419.     Take  the  different  kinds  of  plane  As  and  pass  planes 
through  the  sides  (including  the  sides)  ±  to  the  planes  of  the 
As.     Prove  what  kinds  of  diedral  Z  s  are  formed. 

433. 


Suppose  M  N  1  to  the  intersecting  planes  R  S  and  P  Q. 
How  is  it  related  to  the  line  of  intersection  ? 
Sug.     Erect  a  I  to  the  plane  M  N  at  B.     Show  how  it  is 
related  to  R  S  and  P  Q.     Draw  conclusion. 
Write  Prop.  XIX. 


LINES  AND  PLANES  IN  SPACE. 


259 


434. 
Proposition  XX. 

M 


In  the  figure,  let  A  M  be  the  bisector  of  the  diedral  Z 
C  A  B  D.  From  any  point,  P,  in  A  M,  draw  P  E  1  A  C  and 
P  F  1  B  D.  Pass  a  plane  through  P  E  and  P  F  cutting  A  M 
in  O  P,  A  C  in  E  O,  and  B  D  in  O  F. 

(1)  How  is  plane  P  E  O  F  rftated  to  A  C?  to  B  D?  to  A  B? 

(2)  What  are  the  plane  angles  which  measure  the  two 
equal  diedral  angles?     Prove  it. 

(3)  Compare  P  E,  P  F. 

How  is  any  point  in  the  plane  which  bisects  the 
diedral  Z  located  with  regard  to  the  faces  of  the 
angle  ? 

Exercises. 

420.  What  is  the  locus  of  the  foot  of  an  oblique  line  1  m. 
long  drawn  from  a  point  8  dm.  above  a  plane? 

421.  What  is  the  locus  of  all  points  in  space  equidistant 
from  two  parallel  planes?  from  two  intersecting  planes? 
from  a  given  point?  from  two  given  points? 

422.  What  is  the  locus  of  all  points  in  space  any  given 
distance  from  a  given  plane  ? 


280 


SOLID   GEOMETRY,   BOOK  VI. 


423.  In  the  proposition,  do  P,  B,  O,  F|  lie  in  the  same 
circumference,  or  are  they  concyclic. 

424.  If  two  adjacent  diedral  Z  s  are  supplementary,  show 
how  their  exterior  faces  are  related. 

425.  Pass  a  plane  through  one  of  the  diagonals  of  a 
parallelogram.  Draw  perpendiculars  from  the  extremities  of 
the  other  diagonal  to  the  plane  and  compare  them.  What  is 
the  limit  of  the  length  of  these  is  ? 

426.  If  two  lines  are  perpendicular  to  each  other,  are 
their  projections  always  perpendicular  to  each  other  ? 

427.  Two  parallel  planes  are  intersected  by  two  other 
parallel  planes.  How  do  the  four  lines  of  intersection  compare  ? 

428.  If  O  F  =  F  P,  Prop.  XX.,  how  many  degrees  in 
Z  MABD?     Z  C  B  AD? 

435. 

Proposition  XXI. 

Let  A  B  be  any  straight  line  and  B  C  its  projection  on 
the  plane  M  N. 


How  does  the  acute  angle  formed  by  the  line 
and  its  projection  compare  with  the  angle  formed 
by  the  line  and  any  other  line  in  the  plane  ? 

Sug.  Draw  any  other  line  through  B,  as  B  E.  Measure 
off  B  D  =  B  C. 


LINES  AND  PLANES  IN  SPACE.  261 

Compare  A  C  and  A  D.     Auth. 
Compare  /ABC  and  Z  A  B  D  (§  84). 
Write  the  proposition,  and  call  it  Prop.  XXI. 

436. 

Cor.     With  what  line  in  the  plane  M  N  does  the  line  A  B 
make  the  greatest  angle  ? 

EXERCISK. 


429.  The  projections  of  three  lines  on  the  same  plane 
are  parallel  and  of  equal  length.  Can  you  draw  any  definite 
conclu.sions  concerning  the  three  lines  ? 

437. 

Proposition  XXII. 

Can  you  illustrate  from  the  room  that  two  straight  lines 
in  different  planes  may  have  a  perpendicular  between  them? 

Suppose  A  B  and  C  D  two  straight  lines  not  in  the  same 
plane. 


*•-„ , . 

H                      « 

!       1     : 

^^-- 

^          ^^ 

V "^ 

G^v — A 

\                                            N                                 ^ 

\                                            ^V                              \ 

\                                            ^o                       \ 

\                                                   °                        \ 

(1)  Can  we  find  a  perpendicular  between  the 
lines?     (2)  How  many  perpendiculars  are  there  .^ 

Sug.  Let  P  Q  be  a  plane  passed  through  C  D  and  |1  to 
A  B.  Project  A  B  on  P  Q.  How  does  E  F  compare  with 
A  B?  Is  E  F  II  to  CD?  Why?  Call  their  intersection  G 
Does  A  B  and  its  projection  determine  a  plane?  At  G  erect 
X  G  H  in  the  projecting  plane. 

Pupil  complete  proof.     Can  you  prove  (2)  ?    [§  57.] 


262 


SOLID   GEOMETRY,  BOOK  VI. 


438. 

Cor.    What  is  the  shortest  distance  between  two  lines  not 
in  the  same  plane  ? 

439. 
Proposition  XXIII. 


Given:  The  diedral  Z  s  J  A  B  D,  J'  A'  B'  D'  and  their 
plane  Z  s  C  A  J  and  C  A'  J'. 

To  show  that  the  diedral  Z  s  are  to  each  other 
as  their  plane  Z  s. 

(1)  Suppose  Z  J  A  C  and  J'  A'  Q!  are  commensurable 
and  let  Z  G  A  J  be  the  common  unit.  Express  the  relation 
of  Z  C  A  J  and  Z  C  A'  J'. 

Can  you  pass  planes  through  A  G  and  A  B  ?  A  F  and 
A  B?  A'F'  and  A'  B'?  How  are  the  diedral  Zs  formed 
related?  Can  you  now  show  that  the  diedral  Zs  are  to  each 
other  as  their  plane  Z  s  when  the  plane  Z  s  are  commensurable? 

(2)  Suppose  the  unit  plane  Z  G  A  J  is  contained  twice 
with  a  remainder  in  J"  A''  B''  K.  If  we  let  the  unit  Z  con- 
tinually decrease,  what  does  Z  J"  A"  C  =^  ?  Considering  the 
diedral  Z  s  formed  from  these  plane  /  s  and  decreasing  the 
unit  diedral  Z  indefinitely,  what  does  Y  K"  B''  C"  z^  ? 

Z  J  AC     ,       Z  JABC     , 


What  is  the  ratio 


But 


JAC   _ 

J"  A''  C" 


Z   J 
.  .  . and 


A"  C"-    Z  r  A"B''C' 
diedral   Z  J  A  B  C 


diedral  Z  T  A"  B"  C 


Write  Prop.  XXITT. 


POLYEDRAL   ANGLES. 


263 


POLYEDRAL  ANGLES. 

Definitions. 

440. 

When  three  or  more  planes  meet  in  a  point,  they  form  a 
polyedral  angle  or  polyedral. 

[Thus  the  planes  V  A  B,  V  B  C,  VCD,  V  D  A,  meeting  in  the 
point  V,  form  a  polyedral  angle.    The  point  at  which  the  planes  mee. 


is  called  the  vertex  oi  the  polyedral;  the  intersection  of  the  planes 
are  the  edges;  the  planes  are  called  the  faces;  and  the  angles  A  V  B, 
B  V  C,  etc.,  are  called  th.^  face  angles  of  the  polyedral.] 

Note. — As  in  other  problems,  the  planes  are  of  indefinite 
extent,  but  to  show  the  relation  of  the  edges  in  a  figure  it  is 
clearer  to  have  a  plane  cutting  the  edges. 

441. 

A  polyedral  angle  bounded  by  three  faces  is  called  a 
triedral  angle;  if  bounded  by  four  faces,  it  is  called  a  tetraedral 
angle. 

442. 

In  the  figure  in  §  440,  A  B  C  D  is  called  a  section.  How  is 
it  formed?  If  the  section  is  a  convex  polygon,  the  polyedral  is 
a  convex  polyedral.    What  is  a  coyicave  polyedral? 


264  SOLID   GEOMETRY,   BOOK  VI. 

443. 

The  parts  of  any  polyedral  angle  are  its  face  angles  and 
its  diedral  angles.     Illustrate  with  a  pyramid. 

444. 

The  magnitude  oi  a  polyedral  angle  depends  entirely  upon 
the  divergence  of  its  faces. 

445. 

Two  polyedral  angles  which  have  the  face  angles  and 
diedral  angles  of  one  respectively  equal  to  the  homologous 
face  angles  and 'diedral  angles  of  the  other  and  arranged  in 
the  same  order  are  said  to  be  equal.  •  " 


The  face  angles  A  V  C,  C  V  B,  B  V  A  are  equal,  respect- 
ively, to  A'  V  C,  C  V  B^  B'  V  A',  and  the  diedral  angles 
V  A,  V  C,  V  B,  to  V  A',  V  C,  V  B';  hence  we  may  apply  one 
to  the  other  and  they  will  coincide  in  all  their  parts;  hence 
the  polyedrals  are  equal. 

446. 

Polyedral  angles  which  have  their  face  angles  and  their 
diedral  angles  equal,  each  to  each,  and  arranged  in  reverse 
order,  are  said  to  be  symmetrical. 


FOLYEDRAL  ANGLES. 


265 


The  triedral  angles  V  -  A  B  C  and  V  -  A'  B'  C  are 
symmeiricalM  the  face  angles  AVB,  BVC,  CVA  are  equal, 
respectively,  to  the  face  angles  A'  V  B',  B'  V  C,  Q!  V  A',  and 
the  diedral  angles  V  A,  V  B,  V  C  equal,  respectively,  the 
diedral  angles  V  A',  V  B',  V  C. 

Observe  the  position  of  the  faces. 

[Note. — The  two  hands  or  the  two  feet  or  the  two  sides  of  the 
face  illustrate  symmetrical  solids.  Are  the  right  shoe  and  the  left 
shoe  equal?  What  should  be  said  about  the  right  glove  and  the  left 
glove?] 

447. 

Two  polyedrals  are  vertical  when  the  edges  of  one  are 
the  prolongations  of  the  edges  of  the  other. 

448. 

Proposition  XXIV. 


Can  you  prove  vertical  polyedrals  symmetrical  ? 

\Hi7it. — What  are  the  parts  of  a  polyedral?  Are  the  cor- 
responding vertical  parts  equal  ?  What  is  the  order  of  the 
equal  parts  ?] 

Write  the  proposition,  and  call  it  Prop.  XXIV. 


266 


SOLID   GEOMETRY,  BOOK   VI. 


449. 
Proposition  XXV. 


I,et  S  —  M  R  K  S  be  a  triedral  angle. 

To  show  that  the  two  face  ZsMSR  +  RSK 
are  greater  than  /  M  S  K. 

Sug.  In  M  S  K  draw  S  P  so  that  Z  MSP  =  MSR.  On 
lines  S  R  and  S  P  make  S  D  =  S  B.  Pass  a  plane  through  D 
and  B  cutting  the  solid  Z  in  A  B,  B  C,  A  C.  Compare  A  B, 
A  D;  also  A  B  +  B  C  with  A  D  -^  DC.  Can  you  finish 
proof? 

Write  "Prop.  XXV. 

Question. — Would  Prop.  XXV.  need  proof  if  the  three  face 
angles  were  equal? 


POLYEDRAL  ANGLES. 


287 


450. 
Proposition  XXVI. 


Given:  The  polyedral  ZS—  ABCDEto 
prove  that  the  sum  of  the  plane  Z  s  formed  by  the 
edges  is  less  than  4  rt.  Z  s. 

Siig.  Intersect  the  faces  of  the  polyedral  Z  by  a  plane 
forming  the  plane  figure  A  B  C  D  K.  From  any  point  in  the 
polygon,  as  D,  draw  lines  to  the  vertices,  as  O  B,  O  C,  etc. 
How  many  As  having  vertices  at  O?  How  many  As  with 
vertices  at  S?  What  do  you  observe  in  the  two  sets  of  As? 
What  planes  contain  or  bound  the  triedral  Z  B?  triedral 
Z  C?  etc.     What  did  we  learn  in  Prop.  XXV.?     Can  you  show: 

That  ZSBA-f  ZSBC>  ZABO  +  CBO? 

That  ZSCB+ZSCD>  ZBCO+DCO? 

That  ZSDC+ZSDB>  ZCDO+EDO?  etc. 

How  does  the  sum  of  the  Z  s  at  the  base  of  the  As 
whose  vertices  are  at  S  compare  with  the  sum  of  the  Z  s  at 
the  base  of  the  As  whose  vertices  are  at  O? 


268 


SOLID   GEOMETRY,   BOOK  VI. 


How  many  right  Z  s  are  there  in  each  of  the  two  sets  of 
As  ?  Now  from  these  equal  sums  take  the  sum  of  the  angles 
at  the  bases  of  the  two  sets  of  As.  How  do  the  remainders 
compare?  What  is  the  sum  of  all  the  Z  s  at  O?  What  then 
follows  about  the  sum  of  all  the  angles  at  S? 

Generalize  the  truth  discovered,  and  call  it  Prop.  XXVI. 

451 

Proposition  XXVII. 


Given:  The  two  triedral  Z  s  V— A  B  C,  V,-  A'  B'  C  with 
/  A  V  B  =  Z  A'  V  B';  Z  B  V  C  =  Z  B'  V  C;  Z  C  V  D 
=  Z  C  V  D'. 

Can  you  prove  the  triedrals  equal  or  sym- 
metrical '^ 

Sug,  I.  Compare  the  diedral  Z  s  V  A  and  V  A',  V  B 
and  V  B',  V  C  and  V  C. 

Measure  oflF  equal  distances  from  the  vertices  on  each 
diedral  Z  ,  as  V  A  =  V  B  =  V  C  =  V  A',  etc.  and  compare 
the  As  A  B  C  and  A*  B'  C. 

Sug.  II.  Measure  off  on  V  A  and  V  A'  a  distance  A  D 
—  A'  D'  and  draw  D  E  and  D'  E'  respectively  1  to  A  V  and 
A'  V  in  the  faces  A  V  B  and  A'  V  B';  also  draw  D  F  and  D'  F' 
respectively  1  to  A  V  and  A'  V  in  the  faces  A  V  C  and  A'  V  C 


POLYEDRAL   ANGLES.  269 

Join  the  points  of  intersections  E  and  F,  E'  and  F'.  Can  you 
show  that  D  E  =  D'  E'?  D  F  =:  D'  F'?  E  F  ==  E'  F'?  Com- 
pare Z  E  D  F  and  Z  E'  D'  F'.  Compare  diedral  Z  s  V  A  and 
V'  A'.  Generalize  the  truth  discovered.  Do  you  think  I.  and 
II.  can  be  made  to  coincide?  Suppose  in  III.  V"  A"  ^  V  A 
in  I.,  and  V"  B"  =  V  B,  and  V"  C  =  V  C.  Can  you  make 
them  coincide?     What  do  we  say  of  such  solids  as  I.  and  III.? 

452. 

Cor.  Suppose  two  symmetrical  triedral  Z  s  to  be  isos- 
celes what  follows? 

453. 

A  problem  ot  construction  in  Solid  Geometry  is  consid- 
ered solved  when  it  is  reduced  to  one  of  the  following  element- 
ary constructions. 

(1)  A  straight  line  can  be  drawn  through  any  given  point 
1  to  any  given  plane. 

(2)  A  plane  can  be  passed  through  any  three  given 
points. 

(3)  That  the  intersection  of  a  plane  with  any  other 
plane  or  any  line  can  be  determined. 

In  Solid  Geometry  the  constructions  cannot  be  made  with 
ruler  and  compasses  only. 

Exercises. 

430.  Given  two  straight  lines  t  and  y.  Through  a 
given  point  in  space  determine  a  line  that  shall  cut  the  two 
given  lines. 

431.  A  plane  intersects  2  ||  planes.  Show  (1)  that  the 
alternate  interior  diedral  Z  s  are  equal;  (2)  That  the  cor- 
responding diedral  Z  s  are  equal:  (3)  That  the  sum  of  the 
interior  diedral  Z  s  on  the  same  side  equals  two  rt.  Z  s. 


270  SOLID   GEOMETRY,  BOOK  VI. 

432.  Given  a  plane  to  1  a  line  at  its  middle  point. 
Show  how  any  point  in  the  plane  is  related  to  the  extremities 
of  the  line. 

433.  Can  you  show  that  the  3  planes  which  bisect  the 
diedral  Z  s  of  a  triedral  Z  meet  in  the  same  straight  line  ? 

434.  Find  the  locus  of  all  points,  any  one  of  which  is 
equally  distant  from  two  given  planes. 

435.  Two  II  lines  intersected  by  3  ||  planes  are  12 
inches  and  20  inches  in  length,  If  the  segments  of  the  for- 
mer  are  9  and  3,  what  are   the  segments  of  the  latter. 

436.  Suppose  a  polyedral  Z  formed  by  three  equilat- 
eral Z  s.  What  is  the  sum  of  the  face  angles  at  the  vertex?  If 
formed  by  four?  by  five? 

437.  Can  you  pass  a  plane  through  a  point  jj  to  two 
given  straight  lines  ? 


POLYEDRONS.  271 


BOOK    VII. 

POLYEDRONS. 
Definitions. 

454. 

A  polyedron  is  a  geometric  solid  bounded  by  planes. 

The  intersections  of  the  planes  bounding  the  polyedron 
are  called  the  edges;  the  intersections  of  the  edges  are  called 
vertices-  the  portions  of  the  planes  included  by  the  edges  are 
called  the  faces.     Any  face  may  be  thought  of  as  the  base. 

455. 

Polyedrons  are  classified  as  to  the  number  effaces  required 
to  bound  them. 

\^Question:  What  is  the  fewest  number  of  faces  necessary  to  form 
a  polyedron  ?] 

,  Note. — The  pupil  will  find  that  a  few  cents  invested  in 
putty  or  moulding  clay  will  be  well  spent,  since  with  a  thin- 
bladed  knife  many  concrete  illustrations  of  solids  can  be  read- 
ily made. 

456. 

A  polyedron  of  four  faces  is  called  tetraedron,  one  of  five 
faces  2i  pentaedron,  one  of  six  faces  a  hexaedron,  one  of  eight 


272 


SOLID  GEOMETRY,  BOOK  VII. 


faces  an  octaedron  oue  of  ten  faces  a  decaedron,  one  of  twelve 
faces  a  dodecaedron,  one  of  twenty  faces  an  icosaedron,  etc. 


1 


Tetraedron. 


Hexaedron. 


Dodecaedron. 


Icosaedron, 


POLYEDRONS. 


273 


Scholium.  Having  drawn  on  cardboard  the  diagrams  be- 
low, cut  through  the  heavy  lines  and  half  through  the  dotted 
lines.  By  folding  these  figures  the  regular  polyedrons  can  be 
formed. 


Tttrahedron 


Octahedron 


Hexahedron. 


Dofiecahedron 


Icosahedron 


457. 


A  polyedron  is  cotiuex  when  any  plane  section  is  a  con- 
vex polygon. 

In  the  convex  polyedron  no  face  will  enter  the  polyedron 
when  produced. 

Polyedrons  will  be  considered  convex  in  this  book  unless 
otherwise  stated. 

458. 

A  straight  line  joining  any  two  vertices  not  in  the  same 
face  is  called  a  diaj^onal. 

459. 

The  volume  of  a  solid  is  the  number  expressing  its  ratio 
to  another  solid  arbitrarily  taken  as  the  unit  of  volume. 

The  edge  of  jhe  unit  is  a  linear  unit. 

If  a  cubic  cm.  is  contained  in  a  given  solid  50  times,  its  volume 
is  50  cubic  cm. 

460. 

Two  volumes  are  said  to  be  equivalent -^h^n.  their  volumes 
are  equal. 

18 


27  i  SOLID  GEOMETRY,  BOOK  VII. 

PRISMS. 

461. 

A  prism  is  a  polyedron  two  of  whose  faces,  the  bases,  are 
equal  polygons  having  their  corresponding  sides  parallel,  and 
the  remaining  faces  are  parallelograms  formed  by  planes 
passing  through  the  corresponding  sides  of  the  bases. 

The  parallelograms  are  called  the  lateral  faces. 

Question:     What  form  the  basal  edges?  the  lateral  edges? 

462. 

The  lateral  edges  of  a  prism  ate  parallel  and  equal.  Can 
you  prove  it? 

463. 

A  right  sectioJi  of  a  prism  is  a  section  formed  by  a  plane 
passing  at  right  angles  to  the  lateral  edges. 

464. 

The  altitude  of  a  prism  is  the  perpendicular  distarce  be- 
tween its  bases. 

465. 

Prisms  are  triangular,  quadrangular,  etc.,  according  as 
their  bases  are  triangles,  quadrilaterals,  etc. 

466. 

A  right  prism  is  one  whose  lateral  edges  are  perpendicu- 
lar to  its  faces. 

467. 
An  obliqiie  prism  is  one  whose  lateral  edges  are  oblique  to 
its  faces. 

468. 
A  regular  prism  is  a  right  prism  whose  bases  are  regular 
polygons. 

469. 
The  lateral  faces  of  a  prism  form  a  prismatic  surface.    The 
faces  may  extend  beyond  the  bases. 


PRISMS. 

470. 


275 


A  truncated  prism  is  a  portion  of  a  prism  included  between 
a  base  and  a  plane  not  parallel  to  the  bases  which  cuts  all  the 
iateral  edges. 

471. 

Proposition  I. 


Given:     The  prism  A  B  cut  by  the  parallel  planes  C  F 
and  C'iF'. 


276  SOLID  GEOMETRY,  BOOK  VII. 

Compare  the  polygon  CDEFG  and  C'D^E'F'G'. 

Siig.  Show  how  C  D  and  C  D',  D  E  and  D'  E',  etc.,  are 
related,  and  also  how  Z  C  D  E  and  Z  C  D'  E',  Z  D  E  F  and 
Z  D'  E'  F',  etc.,  are  related. 

Can  you  now  show  the  relation- between  the  polygons? 

472. 

Cor.  (1)  Prove  what  the  section  is  when  the 
prism  is  cut  by  a  plane  ||  to  the  base. 

(2)     How  do  right  sections  compare? 

473. 
Proposition  II. 


Suppose  A  D'  represent  any  oblique  prism  and  F  G  H  I  J 
a  right  section. 

Can  you  find  an  expression  for  its  lateral  surface? 

Sug.     What  are  the  faces?     How  do  the  edges  compare? 
How  does  the  area  of  any  face  compare  with  a  rectangle  hav- 


PRISMS. 


277 


ing  the  same  base  and  an  equal  altitude  ?  Maj-  we  consider  a 
lateral  edge  as  base  of  the  parallelograms  ? 

If,  for  example,  we  take  A  A'  as  the  base  of  the  parallelo- 
gram A'  B,  what  is  its  altitude? 

How  does  the  sum  of  the  altitudes  of  the  faces  compare 
with  the  perimeter  of  a  right  section. 

Complete  proof  and  write  the  generalization.  Call  the 
statement  Prop.  II. 

474. 
Cor,    How  find  the  area  of  a  right  prism? 

475. 
Proposition  III. 


Given-.  In  the  prisms  A  aT,  A'  d'  the  faces  B  E,  B  ^ ,  B « 
respectively  equal  to  B'  E',  BV,  B'a',  and  similarly  arranged 

Compare  the  volumes  of  the  prisms. 

Stig.  Compare  any  two  corresponding  triedral  Z  s,  as  B 
and  B'.  Can  you  make  A  D  coincide  with  A'  D'?  Kb  coincide 
with  A'  /5?  B c  coincide  with  B'  / ?  ab  with  a' b't  be  with  b' e'? 
Show  that  a  d  coincides  with  «'  d' , 


278 


SOLID  GEOMETRY,  BOOK  VII. 


Finish  the  proof  and  write  the  generalization.     Call  this 
Prop.  III. 

476. 

Cor,  L    When  are  two  right  prisms  equal? 

477. 
Cor,  II.    Suppose  the  figures  above  are  truncated 
prisms  and  the  conditions  the  same.    Compare  the 
solids. 

478. 

Proposition  IV. 

Given.     The  oblique  prism  A  D'  and  F  G  H  I  J  a  right 
section. 


Can  you  find  a  right  prism,  having  for  its  base 
a  right  section  Of  the  oblique  prism  and  an  altitude 


PRISMS.  279 

equal  to  a  lateral  edge  of  the  oblique  prism,  which 
shall  be  equivalent  to  the  given  oblique  prism? 

Siig.  Extend  the  lateral  edges  A  A',  B  B',  etc.,  making 
F  F'  =r  A  A'.  At  F'  pass  a  plane  1  F  F'.  How  is  this  plane 
related  to  F  I?  What  is  F  1?  A'  I'?  A  I?  Compare  B  B' 
with  G  G'. 

How  do  B  G  and  B'  G'  campare?  A  B  and  A'  B?  F  G  and 
F'  G'?     What  is  A  G  ?  A'  G?     Compare  them. 

In  a  similar  manner  compare  B  H  and  B'  H'. 

Compare  /ABC  with  /  A'  B'  C  and  base  B  E  with 
base  B'  E'.  Compare  the  triedral  angle  A  B  G  C  with  the 
triedral  A'  B'  G'  C. 

Compare  the  truncated  prisms  A  I  and  A'  I'.  Can  you 
now  complete  the  proof  to  the  answer  of  the  original  question:* 

Write  the  proposition,  and  call  it  Prop.  V. 


PARALLELOPIPEDS. 

Definitions. 

479. 


Kparallelopiped  is  a  prism  whose  bases  are  parallelograms. 

480. 

A  right  parallelopiped  is  one  whose  lateral  edges  are  per- 
pendicular to  the  bases. 


280  SOLID  GEOMETRY,  BOOK  VII. 

481. 

A  rectangular  par allelo piped  is  a  right  parallelepiped  hav- 
ing all  its  faces  rectangles.  How  would  you  define  a  cube? 
(See  Fig.  §  485.) 

HXERCISR. 

438.  Draw  a  right  parallelepiped  whose  bases  are,  (1)  tra- 
peziums, (2)  trapezoids,  (3)  rhomboids,  (4)  rhombuses. 

462. 

PROPOSlTlOlSi    V. 


Given  any  parallelepiped,  A  G,  and  consider  A  H 
and  B  G  the  bases.     Compare  the  opposite  faces. 

[////i/f.— Compare  E  F,  H  G,  D  C,  A  B;  also  B  C,  G  F, 
E  H,  A  D.  Compare  Z  F  E  H  and  Z  B  A  D,  Z  E  H  G  and 
Z  A  D  C.     Compare  faces  A  C  and  E  G.] 

What  can  yon  say  of  the  opposite  faces  of  a  par- 
allelepiped ? 

Write  the  general  statement,  and  call  it  Prop.  V. 
Exercise. 

439.  Let  P  Q  be  a  section  formed  by  passing  a  plane 
through  the  parallelepiped  A  G,  cutting  only  2  pairs  of  op- 
posite sides.     Prove  what  the  section  is. 


PARALLELOPIPEDS.  281 

483. 
Proposition  VI. 


Given  any  parallelepiped,  D  F,  required  to  pass  a  plane, 
through  two  diagonally  opposite  edges. 

What  solids  are  formed?     Compare  them 

\Hi7it. — Let  A  C  G  E  be  the  required  plane.  Pass  a  plane 
M  N  O  P,  1  to  the  edges  of  the  solid,  cutting  the  plane  A  G  in 
NP.     What  is  M  NOP?     PN?] 

Compare  the  oblique  prisms  B  —  F  G  E  with  another 
prism  whose  base  is  N  O  P  with  altitude  A  E. 

Complete  the  demonstration  and  write  the  general  truth. 
Call  it  Prop  VI. 

484. 

Proposition  VII. 

Any  triangular  prism 

Exercises. 

440.  Find  the  entire  surface  of  a  right  triangular  prism 
the  sides  of  whose  base  are  8,  12,  and  16  inches  and  altitude 
30  inches. 

441.  In  how  many  ways  can  a  polyedral  Z  be  formed 
with  equilateral  As?  with  squares?  with  regular  pentagons.^ 
with  a  regular  hexagon  ?  with  regular  heptagons  ? 


282  SOLID  GEOMETRY,  BOOK  VII, 

442.  What  is  the  entire  surface  of  a  regular  hexagonal 
prism,  each  side  of  the  base  being  10  centimeters  and  the  alti- 
tude 20  centimeters. 

443.  The  four  diagonals  of  a  parallelepiped  bisect  each 
other. 

444.  The  diagonals  of  a  rectangular  parallelepiped  are 
equal. 

445.  The  diagonals  of  an  oblique  parallelepiped  are 
unequal. 

446.  How  many  sets  of  ||  lines  in  a  parallelepiped  ? 

447.  In  any  parallelepiped  the  sum  of  the  squares  on 
the  4  diagonals  equals  the  sum  of  the  squares  on  the  12 
edges. 

448.  Given  a  rectangular  parallelepiped  to  show  that  if 
the  diagonals  of  all  the  faces  are  drawn  and  the  points  of  in- 
tersection of  the  diagonals  of  the  opposite  faces  are  connected, 
these  connecting  lines  are  concurrent  at  the  middle  point  of 
each. 

449.  If  a  plane  cuts  one  edge  of  a  prismatic  space,  it 
cuts  all  the  edges.  What  follows  if  it  cuts  one  edge  per- 
pendicularly ? 

450.  What  should  be  the  edge  of  a  cube  so  that  its  entire 
surface  shall  be  a  square  foot  ? 

451.  A  rectangular  parallelepiped  is  x,  \yy  y,  by  z.  What 
is  the  longest  straight  line  that  can  be  drawn  in  the  solid  ? 

452.  Show  what  every  section  of  a  prism  is  when  a  plane 
cuts  it  parallel  to  the  base. 

453.  Construct  a  pentagonal  prism  and  shew  what  is  the 
sum  of  the  plane  angles  of  the  lateral  diedral  angles. 

454.  In  the  last  exercise  how  many  face  angles  f  diedral 
angles  ?  triedral  angles  ?  If  the  base  have  r  sides  instead  of 
£ve,  hew  many  face  angles?  etc.? 

455.  Given  a  cube  with  an  8-inch  edge.  Join  the  mid- 
dle point  of  each  face  with  the  other  middle  points.  What  are 
the  plane  figures  formed  ?     Can  you  compare  the  sum  of  their 


PARALLELOPIPEDS. 


283 


surfaces  with  the  surface  of  the  original  cube?  What  is  the 
solid  figure  formed  ? 

456.  Given  a  rectangular  parallelopiped  whose  base  is  6 
by  8  and  whose  altitude  is  12.  Join  the  middle  points 
of  each  face  to  the  middle  points  of  the  others.  What  are 
the  plane  figures  formed  ?  Can  you  compare  the  sum  of 
their  surfaces  with  the  surface  of  the  parallelopiped  ?  What  is 
the  solid  figure  formed  ? 

Note. — The  last  two  exercises  illustrate  two  forms  of  crystals 
sometimes  seen  in  the  mineral  kingdom. 


485. 


Proposition  VIIT. 


Can  you  show  that  two  rectangular  parallelo- 
pipeds  having  equal  bases  are  to  each  other  as  their 
altitudes  when  the  altitudes  are  commensurable? 


2841 


SOLID  GEOMETRY,  BOOK  VII. 


Compare  the  two  parallelopipeds  when  the  alti- 
tudes are  incommensurable. 

Sy^.  Let  P  and  Q  be  the  solids  and  suppose  m  n  and  rt 
to  be  commensurable,  /  s  being  less  than  the  unit  of  measure. 
Call  the  lower  parallelopiped  formed  by  passing  a  plane 
through  /  II  to  the  base,  Q^  How  are  PandQ'  related?  Write 
the  relation.  Is  it  always  true  for  any  unit  parallelopiped? 
Now   suppose   the  linear  unit  on  m  n  to  be  indefinitely   de- 

creased,  how  will  it  affect  r  tf     Q?   Write  -~~}  ^,z^'> 

Therefore Generalize. 

Call  this  Prop.  VIII. 

Exercise. 
457.     Explain  the  statement  that  if  two  rectangular  par- 
allelopipeds have  two  dimensions  in  common,  they  are  to  each 
other  as  their  third  dimensions. 


PARALLELOPIPEDS. 


285 


486. 
Proposition  IX. 


{ 


f 

R 

^ 

1 
1 

\ 

\ 

\     1 

Given:  The  two  rectangular  parallelepipeds  P  and  Q 
having  two  dimensions,  b  and  c,  equal.  Write  an  expression 
to  show  the  relation  of  P  and  Q.  Call  this  equation  (I).  Next 
let  Q  and  R  be  two  rectangular  parallelepipeds  having  two 
dimensions,  a  and  c,  equal.  Write  a  second  equation  showing 
their  relation.     Multiply  (1)  by  (2)  and  simplify. 

What  is  the  meauing  of  the  resulting  equation? 
Can  you  write  the  proposition  deduced  ? 
Call  it  Prop.  IX. 

Exercise. 

458.  If  P  in  §480  have  edges  3,  4,  5  cm.,  what  is  the 
area  of  a  diagonal  plane  ? 


286 


SOLID  GEOMETRY,  BOOK  VII. 
487. 

Proposition  X. 


-\ 


Compare  the  rectangular  parallelopipeds  P  and 
R,  whose  edges  are  x^  y,  z  and  x\  y\  z\ 

Sug.  Construct  a  third  parallelepiped,  Q,  with  edges 
x' ,  z\y.  Write  an  equation  comparing  P  and  Q.  Call  it  (I). 
In  the  same  manner  compare  Q  and  R.  Call  this  equation 
(2).  Multiply  (1)  by  (2)  and  simplify.  Explain  the  meaning 
of  this  third  equation.  Generalize. 
■    Call  this  Prop.  X. 

488. 

Cor.     Suppose  that   P  were  a  cube   with  edges  J,  1,  1 
Write  the  equation  expressing  the  relation  of  R  to  P. 


If  we  call  P   a   unit  of  volume,  ^\\2it  diO&s  your   equation 
show  as  to  the  number  of  units  of  volume  iu  R  ? 


PARALLELOPIPEDS. 


287 


What  expresses  the  volume  of  a  rectangular  par- 
allelopiped? 

ExERCfSE. 

459.  When  the  edges  of  the  rectangular  parallelepiped 
are  multiples  of  the  linear  unit,  construct  a  figure  showing 
how  to  compute  the  volume  of  the  solid. 

489. 

Proposition  XI. 


Given  any  oblique  paiallelopiped,  A  G;  i.  e.,  all  angles 
oblique. 

Produce  the  edges  A  B,  E  F,  D  C,  H  G. 

Make  E'   F'  =  E  F.     Pass  planes  E'  D',  F'  C  1  to  E'  F'. 

Sicg.  1 .  By  considering  E'  D'  and  F'  C  bases,  what  kind 
of  a  solid  have  we?-    [i^478.] 

How  does  the  edge  E'  F'  compare  with  E  F? 

Compare  E  C  and  E'  C     [§  478.  ] 

Sicg.  2.  Now  consider  D'  C  G'  H'  as  the  base  of  A'G'. 
Produce  D'  A',  making  A'  M  =:  D'  A'.  Produce  H'  E',  G'  F', 
C  B',  making  B'  N  :=  C  B'.  Make  right  section  A'  B'  J  I 
and  M  N  L  K.  What  kind  of  solid  is  A'  L  ?  Compare  A'  L 
with  A' G'.     [Prop,  v.,  §478.] 


288  SOLID  GEOMETRY,  BOOK  VII. 

How  does  it  compare  with  AG? 
Can  you  generalize  the  result? 
Call  this  Prop.  XI. 

490. 
Cor.  I.     How  do  find  the  volume  of  any  paral- 
lelopiped? 

491. 
Cor,  II.     Construct   a  figure    and  show  how  to 
find  the  volume  of  any  triangular  prism. 

492. 

Proposition  XII. 

Given  a  prism  whose  base  is  a  polygon  of  7i  sides.  Into 
how  many  triangular  prisms  may  it  be  divided  by  passing 
planes  through  the  lateral  edges? 

Construct  a  figure  and  show  how  to  find  the 
volume  of  any  prism  ? 

Write  the  general  truth,  and  call  it  Prop.  XII. 

493. 

Cor.  Can  you  show  that  any  two  prisms  are 
to  each  other  as  the  products  of  their  bases  and 
altitudes  ? 

Compare  two  prisms  having  equivalent  bases.  Compare 
two  prisms  having  =  altitudes. 

In  figure  Prop.  XL  may  we  compare  A  G  and  A'  G'  as 
prisms?  A  G  and  A'  L  ?  What  statement  can  you  make  con- 
cerning these  prisms?  of  any  two  prisms  having  equivalent 
bases  and  equal  altitudes? 

Exercises. 
460.     The  dimensions  of  the  base  of  a  rectangular  par- 
allelopiped  are  3  and  4  centimeters  and  the  entire  surface  is  52 
square  centimeters.     Find  the  volume. 


PYRAMIDS. 


289 


461.  The  volume  of  a  rectangular  parallelopiped   is  60 
cubic  centimeters,  the   entire  surface  94   square  centimeters, 
and    the  altitude  is  3  centimeters.    Find  the  dimensions   of 
the  base. 

462.  Find  the  lateral  surface  of  a  regular  triangular 
prism,  each  side  of  whose  base  is  5  centimeters,  and  whose 
altitude  is  10  centimeters.  Suppose  the  base  were  6,  5,  5  cen- 
timeters, what  would  be  the  entire  surface? 

463.  Find  convex  surface  and  volume  of  a  regular  hex- 
agonal prism  each  side  of  whose  base  is  1  dm.  and  whose  alti- 
tude is  10  dm. 

464.  Two  triangular  prisms  P  and  Q,  have  the  same  alti- 
tude; P  has  for  its  base  a  right  isosceles  triangle;  Q  has  for  its 
bise  an  equilateral  triangle  of  side  equal  to  the  hypotenuse 
of  the  base  of  P.    What  is  the  ratio  of  the  volumes  of  P  and  Q  ? 

465.  Find  the  ratio  of  the  conv-ex  surfaces  of  P  and  Q,  in 
Ex.  464. 

466.  A  rectangular  parallelopiped  is  4,6,  and  9  centi- 
meters.   What  is  the  edge  of  an  equivalent  cube  ? 


PYRAMIDS. 


494. 

A  pyramid  is  a  polyedron  bounded  by  a  polygon,  and  a 
series  of  triangles  which  meet  in  a  common  point  and  whose 
bases  are  the  sides  of  the  polygon. 

Thus  the  polygon  A  B  C  D  E  is  called  the  baseoi  the  pyra- 


mid.    The  common  vertex  of  the   triangular   faces  is  called 


290 


SOLID  GEOMETRY,  BOOK  VII. 


the  vertex  of  the  pyramid.  The  edges  passing  through  the 
vertex  are  called  the  lateral  edges.  The  perpendicular  from 
the  vertex  to  the  base  is  called  the  altitude. 

495. 

A  pyramid  is  triangular,  quadrangular,  pentagonal,  etc., 
according  as  its  base  is  a  triangle,  a  quadrilateral,  a  penta- 
gon, etc. 

496. 


A  regular  pyramid  has  for  its  base  a  regular  polygon,  and 
its  vertex  lies  in  the  perpendicular  erected  at  the  center  of  the 
polygon. 

497. 
Proposition  XIII. 


Can  you  show  how  the  lateral  edges  of  a  regular 
pyramid  are  related  ? 


PYRAMIDS.  291 

498. 

Cor.  /.  What  kind  of  triangles  are  the  lateral 
faces  of  a  regular  pyramid  ? 

499. 

Cor.  II.  How  do  the  altitudes  of  the  lateral 
faces  drawn  from  the  common  vertex,  V,  of  a  regu- 
lar pyramid  compare? 

500. 

The  altitude  of  any  of  the  lateral  faces  of  a  regular  pyra- 
mid drawn  from  the  common  vertex  is  called  the  slant  height. 

501. 

The  lateral  surface  of  a  pyramid  is  the  sum  of  the  areas 
of  its  lateral  faces, 

502. 

A  truncated  Pyramid  is  the  portion  of  a  pyramid  included 
between  its  base  and  a  plane  cutting  all  the  lateral  edges. 
[1 470.] 

503. 

A  frustum  of  a  pyramid  is  a  truncated  pyramid  whose 
bases  are  parallel. 


The  altitude  of  a  frustum  is  the  perpendicular  distance 
between  the  planes  of  its  bases. 

The  slant  height  of  a  frustrum  of  a  regular  pyramid  is 
the  altitude  of  any  lateral  face. 


292  SOLID  GEOMETRY,  BOOK  VII. 

Exercises. 

487.     What  are  the  faces  of  the  frustum  of  a  regular 
pyramid  ?    Can  you  prove  that  they  are  equal  ? 

468.  Can  you  find  an  expression  for  the  lateral  sur- 
face of  the  frustum  of  a  pyramid  ? 

469.  Can  you  find  an  expression  for  the  lateral  area  of 
any  regular  pyramid? 

470.  Can  you  draw  a  figure  illustrating  geometrically  the 
formula  (x  -\-j/)  ^  =  x^  -\-  Sx^jy -{■  Sxy^  -\-  xy^} 

471.  A  rectangular  drain-tile  has  the  dimensions  60  cm., 
12  cm.,  10  cm.  The  mold  is  65  cm.  long  and  proportionally 
wide  and  deep.  Supposing  there  were  no  orifice,  what  per 
cent  does  the  tile  decrease  in  baking  ? 

504. 
Proposition  ;^IV. 


Given:  Any  oblique  pyramid,  asS  —  ABCDE,  cut  by 
a  plane  ||  to  its  base  intersecting  the  edges  \\i  a  b  c  d  e  and  the 
altitude,  S  O,  in  o. 

1.  Can  you  show  how  the  edges  and  altitudes 
are  divided  ? 

Sug,  How  are  a  b  and  A  B  related  ?  Why  t  b  c  and 
BC?etc. 


PYRAMIDS. 


293 


2.  Compare  the  section  of  the  pyramid  with 
the  base. 

Sug.  Compare  Z  abc  and  Z  ABC,  Z  b  c  d  2irA  / 
BCD,  etc. 

What  can  you  say  of  the  two  polygons  ? 

Can  you  prove  the  homologous  sides  proportional  ? 

What  results? 

Put  the  answers  to  1  and  2  into  a  general  statement,  and 
call  it  Prop.  XIV. 

505. 

■** 

Cor,  /.  Can  yon  show  that  any  section  of  a 
pyramid  ||  to  the  base  is  to  the  base  as  the  square  of 
its  distance  from  the  vertex  is  to  the  square  of  the 
altitude  of  the  pyramid  .'^ 

Sug.  Compare  the  two  polygons  with  two  homologous 
sides,  the  segments  of  an  edge,  etc. 

506. 


Cor.  II.     (1)  Suppose  S  — A  B  C  D  and  P  —  Q  R  S  to  be 
two  pyramids  of  equal  altitudes.     Let  each  pyramid  be  cut 


294 


SOLID  GEOMETRY,  BOOK  VII. 


by  a  plane  ||  to  its  base  inad  c  dTand  ^  rs,  respectively,  and  let 
each  plane  be  equally  distant  from  the  vertex. 

Compare  the  ratio  of  the  sections  -' ith  the  ratio 
of  the  bases. 

Altitude  S  I  =  altitude  P  w,  and  altituc      *^      _::;  altitude 

Sug.    Use  Cor.  I. 

(2)     Suppose  the  base  of  S  —  A  B  C  D   equivalent    to 
base  of  P  —  Q  R  S,  what  deduction  follows  from  (1). 
Write  Cor.  II. 

507. 

Proposition  XIV. 


If  in  any  pyramid  we  inscribe  and  circumscribe 
a  series  of  prisms  of  equal  altitudes,  what  is  the 
limit  of  the  sum  of  each  series  when  the  altitude  is 
indefinitely  diminished  ? 

Sng.     Given  pyramid  V  —  ABC  with  altitude  A  R. 
Suppose  the  altitude  divided  into  any  number  of  equal 
parts,  letting  uhe  a.  unit  of  measure.     Pass  planes  through 


PYRAMIDS.  295 

these  points  of  division  parallel  to  the  base,  cutting  the  pyra- 
mid at  L,  nt,  and  n.  What  will  the  sections  be?  Upon  ABC 
and  the  sections  at  L,  ni,  and  n  as  lower  bases,  construct 
prisms  having  altitudes  equal  to  u  and  the  lateral  edges  paral- 
lel to  A  V.  These  are  the  circumscribed  prisms.  Now  with 
the  sections  at  L,  m,  and  n  as  upper  bases  and  altitudes  equal 
to  u  and  lateral  edges  parallel  to  A  V,  construct  prisms.  These 
are  the  inscribed  prisms. 

How  does  the  sum  of  two  inscribed  prisms  in  any  pyra- 
mid compare  with  the  sum  of  5  inscribed  prisms  ?  with  10  ? 
with  100  ? 

How  does  the  sum  of  3  circumscribed  prisms  of  any  pyr- 
amid compare  with  the  sum  of  5  circumscribed  prisms?  with 
10?  with  100?  (It  is  very  important  that  the  pupil  master 
these  questions  before  proceeding.)  If  the  altitude  of  each 
series  of  prisms  be  indefinitely  decreased,  how  does  it  affect 
the  sum  of  each  series  ? 

What  is  the  limit  of  the  sum  of  the  series  of  inscribed 
prisms  ?  of  the  sum  of  the  circumscribed  series  ? 

Write  a  general  statement,  and  call  it  Prop.  XIV. 

Exercises. 

472.  Each  face  of  a  triangular  pyramid  is  an  equilateral 
triangle  whose  side  is  4  cm.     Find  the  surface. 

473.  The  side  of  the  base  of  a  square  pyramid  is  4  m.,  the 
altitude  is  10  m.;  how  many  square  meters  of  tin  is  required  to 
cover  it  ? 

474.  The  diagonal  of  the  face  of  a  cube  is  a  \I~2.  Find 
the  volume. 

475.  Can  you  cut  a  cube  with  a  plane  so  that  the  section 
shall  be  a  regular  hexagon  ? 

476.  Can  you  show  that  the  lateral  area  of  any  pyramid 
is  greater  than  the  area  of  its  base  ? 


296 


SOLID  GEOMETRY,  BOOK  VII. 

508. 

Proposition  XV. 


Given  any  two  pyramids,  as  V  —  A  B  C,  V  —  A'  B'  C, 
having  equivalent  bases  in  the  same  plane  and  equal  altitudes, 
A  R  =  A'  R^ 

Compare  the  volumes  of  the  pyramids. 

Sug.  1.  Divide  A  R  into  any  number  of  equal  parts,  let- 
ting u  be  one  of  the  equal  parts.  Pass  planes  through  the 
points  of  division  parallel  to  the  bases.  Compare  correspond- 
ing sections. 

Sug.  2.  Inscribe  a  series  of  prisms  in  each  pyramid 
having  the  sections  as  upper  bases  and  common  altitude  u. 
Compare  any  two  corresponding  prisms.  Compare  the  total 
vSum  of  the  prisms,  P,  in  V  —  ABC  with  the  total  sum  of  the 
prisms,  P',  in  \'  —  A'  B'  C.  Now  suppose  u  be  indefinitely 
decreased,  how  does  it  aflfect  the  prisms?  What  does  P  = 
P'=? 

Complete  the  work,  and  write  Prop.  XV. 


PYRAMIDS. 


297 


509. 

Cor.  I.  Suppose  a  pyramid  has  a  parallelogram  for  its 
base  and  a  plane  be  passed  through  two  opposite  edges,  cut- 
ting the  base.     What  follows  ? 

510. 

Cor.  II.  Given  a  pyramid  having  a  parallelogram  for  its 
base  which  is  double  the  area  of  the  base  of  a  triangular  pyr- 
amid of  the  same  altitude.     How  do  they  compare? 

Exercise. 
477.     How  may  the  volume  of  any  polyedron  be  found? 

511. 

Proposition  XVI. 


Given:     The  triangular  prism  O  —  ABC. 
Prove  that  it  may  be  divided  into  three  equiva- 
lent pyramids. 


298  SOLID  GEOMETRY.  BOOK  VII. 

Stij^.  1 .     Pass  a  plane  through  O  and  A  C. 

How  is  the  prism  divided  ? 

Prove  that  the  quadrangular  pyramid  may  be  divided  into 
two  equal  pyramids. 

Sug.  2.  Compare  C  -  D  E  O  with  O  -  D  E  C,  with  O  — 
ABC. 

Finish  the  proof  and  write  the  general  statement.  Call 
this  Prop.  XVI. 

512. 

Cor.  /.  Compare  the  volume  of  a  triangular 
prism  with  that  of  a  pyramid  having  the  same  base 
and  altitude. 

513. 

Cor.  II.  Compare  a  triangular  pyramid  with  a 
prism  having  an  equivalent  base  and  altitude. 

514. 

Cor.  III.  How  find  the  volume  of  a  triangular 
pyramid? 

515. 

Cor.  IV.  Show  that  any  pyramid  may  be  di- 
vided into  triangular  pyramids.    How  many.^ 

516. 

Cor.  V.  Show  how  pyramids  of  equivalent  bases 
are  related.  Suppose  two  pyramids  have  equal  alti- 
tudes, show  how  they  are  related.  Suppose  two  pyr- 
amids have  equivalent  bases  and  equal  altitudes, 
show  how  they  are  related. 


PYRAMIDS. 


299 


517. 
Proposition  XVII. 


Given:  Any  two  tetraedrons,  as  V  —  A  B  C  and  V  — 
A'  B'  C,  having  the  triedral  Z  A  equal  to  the  triedral  Z  A'. 

Prove  that  the  tetraedrons  are  to  each  other  as 
the  products  of  the  edges  of  the  corresponding  trie^ 
dral  zs. 

Sug.  Apply  the  smaller  tetraedron  to  the  larger,  so  that 
Z  A'  coincides  with  A.  Drop  1  s  V  P,  V  P'  to  the  bases  ABC 
and  A'  B'  C  (Are  the  two  bases  in  the  same  plane  ?)  Can  you 
pass  a  plane  through  V  P  and  V  P?  Will  it  pass  through  V  V? 
through  A?  What  is  its  line  of  intersection  with  ABC? 
How  are  two  triangular  prisms  related  ?  two  triangular  pyr- 
amids? two  tetraedrons?  Call  the  tetraedrons  T  and  T'. 
(1)T:T'::  ABC  —  :A'B'C'  — .  (2)  ABC:  A'B'C  :  :  — :— . 
How  are  V  P  and  V  P'  related?  Substitute  and  finish  the 
proof. 

Write  the  proposition,  and  call  it  Prop.  XVII. 


300 


SOLID  GEOMETRY,  BOOK  VII. 


Question,  If  V  —  A  B  C  and  V  —  A'  B'  0!  are 
similar  tetraedrons,  can  you  show  that  they  are  to 
each  Other  as  the  cubes  of  their  homologous  edges  ? 

•  518. 

Proposition  XVIII. 


Construct  a  figure  and  show  how  to  find  the  volume  of 
any  pyramid.    (§511.) 

Write  the  proposition,  and  call  it  Prop.  XVIII. 
Given:     The  frustum  of  any  triangular  pyramid. 

Can  you  divide  the  frustum  into  three  pyramids, 
one  having  for  its  base  the  lower  base  of  the  frustum, 
another  having  for  its  base  the  upper  base  of  the 
frustum,  and  the  third  having  a  base  equivalent  to  a 
mean  between  the  upper  and  lower  bases  of  the  frus- 
tum, and  each  pyramid  to  have  the  altitude  of  the 
frustum  of  the  pyramid  ? 

Sug,  1.  PassaplanethroughB',  ACandB',  A',  C.  Con- 
sider B'  -  A  B  C  and  B'—  A  A'  C  as  having  a  common  vertex,  C. 


PYRAMIDS.  301 

Write  an  equation  showing  the  relation  between  C  —  A  B  B' 
and  C  — AB'A'.     Call  this  (1). 

What  do  you  notice  about  the  altitudes  of  the  bases  of 
these  pyramids?  Write  an  equation  showing  the  relation 
birtween  the  two  triangles.     Call  it  (2), 

Substitute  in  (1).  New  equation  (3).  How  then  is  B'  — 
A  B  C  and  B  —  A  A'  C  related  ? 

S?ig.  2.  Write  an  equation  showing  the  relation  between 
B'  — A  A'  C  and  B' —  A'  C  C.  Call  this  (4).  Compare  A^- 
A  A'  C  and  A  C  C.  Call  this  (5).  Substitute  (5)  in  (4).  New 
equation  (6). 

How  does  A'  B'  C  compare  with  ABC?     [§508.] 

Sug.  3.     Can  you  show  that  A  B  :  A'  B' :  :  A  C  :  A'  C  ? 

Form  a  new  proportion  from  this  by  substituting  pyra- 
mids. What  is  B' — A  A' C  from  this  proportion  ?  Call  the 
altitude  of  the  frustum  A,  area  of  lower  base  B,  area  of  upper 
base  d.  What  expresses  the  volume  of  B' — ABC?  Of 
C  -  A'  B'  C  ?  or  B'  —  A'  C  C  ?  Express  B'  —  A'  A  C  in  terms 
of  B,  d,  h. 

Write  a  generalization  of  the  truth  developed,  and  call  it 
Prop.  XVIII. 

519. 

Cor.     (1)  Let  V=volume  of  frustum.     [§518.] 

Prove  V  =  4(B-h/^+  ^  B  x  ^). 
(2)     From  (1)  can  you  show  that 

V  =g(B  -f-  <5  +  v^B  X  b)  is  true  for  the  frustum 

of  a7iy  pyramid  ? 

Siig.  Construct  a  triangular  frustum  in  the  same  plane, 
having  a  lower  base  equivalent  to  the  lower  base  of  the  given 
frustum  and  of  equal  altitude. 


302 


SOLID  GEOMETRY,  BOOK  VII. 

520. 
Proposition  XIX. 


Given:     ABC —  DEC,  any  truncated  triangular  prism. 

Can  you  show  that  the  trtmcated  prism  is  equiv- 
alent to  three  pyramids  having  a  common  base,  ABC, 
with  the  vertices  at  D,  E,  C'.^ 

Sug.  1.  Pass  a  plane  through  E,  A,  C,  and  D,  B,  C,  and 
C',  A,  B.  Can  you  take  E  as  a  vertex  and  name  three  pyramids 
that  compose  the  truncated  prism  ?  Compare  E  —  ADC  and 
B  —  AD  C.  Compare  B  —  A  D  C  and  D  —  A  B  C.  What 
then  is  the  equivalent  of  E  —  ADC? 

Sug.  2.  Compare  A  D  C  C  and  A  C  C.  Compare 
E  —  A  C  C  and  B  —  A  C  C  ?  Also  B  -  A  C  C  with  Q!  — 
ABC.  What  is  the  equivalent  ofE— ACC?  .-.ABC- 
DEC  is  equivalent  to  — ,  — ,  — ,  — . 

Write  the  generalization  of  the  truth  developed,  and  call  it 
Prop.  XIX. 

521. 

Cor,  L  Show  that  the  volume  of  any  truncated 
right  triangular  prism  equals  the  product  of  its  base 
by  \  of  the  sum  of  its  lateral  edges. 


SIMILAR  POLYEDRONS. 


303 


522. 

Cor.  II.  Show  that  the  volume  of  any  trun- 
cated triangular  prism  equals  the  product  of  a  right 
section  by  \  the  sum  of  its  lateral  edges. 

Sug.  After  passing  a  plane  at  right  angles  to  the  edges, 
compute  each  part  separately. 


SIMILAR  POLYEDRONS 
523. 

Definition. 

Similar  polyedrons  are  those  polyedrons  having  the  same 
number  of  faces,  similar  each  to  each  and  similarly  placed, 
and  their  homologous  polyedral  angles  equal. 

The  faces,  lines,  and  angles  of  similar  polyedrons  which 

are  similarly  placed  are  called    liovwlogoiis  laceSy   lines,  and 

angles. 

524. 

Proposition  XX. 


Given  :     V  —  A  B  C  and  V  —  A'  B'  C,  similar   poly- 
edrons. 

Compare  their  homologous  edges. 


304  SOLID  GEOMETRY,  BOOK  VII. 

Sug.  Can  V  —  A'  B'  C  be  made  to  coincide  with  any 
part  of  V  — ABC? 

Generalize.     Call  it  Prop.  XX. 

525. 

Cor.  I.  Can  you  show  tliat  any  two  homologous 
faces  of  two  similar  polyedrons  are  proportional  to 
the  squares  of  any  two  homologous  edges? 

526. 

Cor.  II.  Can  you  show  that  the  entire  surfaces 
of  any  two  similar  polyedrons  are  proportional  to  the 
squares  of  any  two  homologous  edges. 

527. 

Cor,  III.  How  do  the  homologous  diedral  an- 
gles of  similar  polyedrons  compare  ? 

528. 

Proposition  XXI. 

If  a  tetraedron    is  cut  by  a  plane  parallel  to  one  of  the 
faces,  show  that  the  tetraedron  cut  off  is  similar  to  the  first. 
Sug.     Use  figure  in  Prop.  XX. 
Call  this  truth  Prop.  XXI. 

529. 

Proposition  XXII. 

In  figure  for  Prop.  XX.  let  V  —  A  B  C  and  V  —  A'  B'  Q! 
be  two  tetraedrons  with  the  diedral  angles  V  A  and  V  A' 
equal,  and  the  faces  V  A  B  and  V  A'  B'  similar  and  VAC 
similar  to  V  A'  C,  and  these  pairs  of  faces  similarly  placed. 

Can  you  prove  the  tetraedrons  similar? 

Sug.     Suppose  V  —  A'  B'  C  on  V  —  A  B  C. 

Write   the  general  proposition,  and  call  it  Prop.  XXII. 


SIMILAR  POLYEDRONS. 


305 


530. 

Proposition  XXIII. 


Given:  The  two  similar  polyedrons  B  —  D  F  E  and 
H  -  N  O  M,  in  which  triedral  Z  H  =  Iriedral  Z  C,  etc. 

To  Prove — That  they  may  be  decomposed  into 
the  same  number  of  tetraedrons  similar  each  to 
each  and  similarly  placed. 

Sit^.  Draw  the  diagonals  A  F,  C  F.  D  C,  G  O,  K  O, 
K  N.  Can  you  show  that  the  tetraedron  F  —  A  B  C  is  the 
corresponding  tetraedron  to  O  —  G  H  K  ?  Prove  them  simi- 
lar.    Complete  the  proof. 

Write   the  general  truth,  and  number  it  Prop.  XXIII. 

531. 

Proposition  XXIV. 

Given:  F  —  ABC  and  O  —  G  H  K,  two  polyedrons 
composed  of  the  same  number  of  tetraedrons  similar  each  to 
each  and  similarly  placed.    (See  figure  under  Prop.  XXIII.) 

To  Prove— ^  —  ABC  similar  to  O  —  G  H  K. 


20 


306 


SOLID  GEOMETRY,  BOOK  VII. 


Sug.  Can  you  show  how  A  B  F  D  and  G  H  O  N  are  re- 
lated, etc.  ?  Can  you  show  how  the  polyedral  Z  at  A  is  re- 
lated to  the  polyedral  Z  at  G  ?     Complete  proof. 

Draw  conclusion,  and  call  this  Prop.  XXIV. 

532. 

Proposition  XXV. 


c  c- 

Given:  A  —  B  C  I)  and  A'  —  B'  C  D',  two  similar  tetra- 
edrons. 

To  Prove — That  they  are  to  each  other  as  the 
cubes  of  their  homologous  edges. 

Sug.     See  §  517. 

Write  the  proposition,  and  call  it  Prop.  XXV. 

533. 

Cor.  Show  that  any  two  similar  polyedrous  are 
to  each  other  as  the  cubes  of  their  homologous  edges. 

EXKRCISES. 

478.  In  Ex.  455,  compare  the  volume  of  the  solid 
formed  with  the  original  solid. 

479.  In  Ex.  455,  compare  the  volume  of  the  solid 
formed  with  the  original  solid. 


EXERCISES.  307 

480.  A  farmer  wishes  to  build  a  cubical  bin  that  will 
hold  100  bushels  of  wheat.  What  will  be  an  inside  edge  in 
inches  ? 

481.  What  is  the  edge  of  a  cube  whose  entire  surface  is 
1  square  foot? 

482.  What  is  the  entire  surface  of  a  common  building 
brick  ? 

483.  What  is  the  edge  of  a  cube  that  will  contain  a  gal- 
lon, dry  measure? 

484.  The  base  of  a  pyramid  is  12  square  feet  and  its  al- 
titude is  6  feet.  What  is  the  area  of  a  section  parallel  to  the 
base  and  2  feet  from  it  ? 

485.,  Prove  the  diagonals  of  a  rectangular  parallelopiped 
equal. 

486.  The  volume  of  any  triangular  prism  equals  one 
half  the  product  of  any  lateral  face  by  its  distance  from  the 
opposite  edge.     Prove. 

487.  Show  that  the  four  diagonals  of  a  parallelopiped 
bisect  each  other.  (The  point  of  intersection  is  called  the 
center  of  the  parallelopiped.) 

488.  Can  you  prove  that  a  straight  line  passing  through 
the  center  of  a  parallelopiped  and  terminated  by  two  faces 
is  bisected  at  the  center? 

489.  Can  you  show  that  the  middle  points  of  the  edges 
of  a  regular  tetraedron  are  the  vertices  of  a  regular  octaedron. 

Is  the  altitude  of  a  regular  tetraedron  equal  to  the  sum 
of  the  perpendiculars  to  the  faces  from  any  point  within  the 
figure  ? 

490.  Find  the  volume  of  a  regular  triangular  pyramid 
whose  basal  edge  is  4  feet  and  whose  altitude  is  v^y'feet. 

491.  Find  the  lateral  edge,  lateral  area,  and  volume 
of  a  frustum  of  a  regular  triangular  pyramid  the  sides  of 
whose  bases  are  10  v^3'  and  2  v^y"  and  whose  altitude  is  10. 

492.  If  the  homologous  edges  of  two  similar  polyedrons 
are  2  and  3,  what  is  the  ratio  of  their  entire  surfaces  and  of 
their  volumes? 


308  SOLID  GEOMETRY,  BOOK  VII. 

493.  Can  you  show  that  the  volume  of  a  regular  tetra- 
edron  equals  the  cube  of  an  edge  multiplied  ^Y  j^r   V^?? 

494.  Can  you  show  that  the  volume  of  a  regular  octa- 
edron  equals  the  cube  of  an  edge  multiplied  by  -q-   v^^? 

495.  A  side  of  a  cube  is  the  base  of  a  pyramid  whose 
vertex  is  at  the  center  of  the  cube.  Compare  the  volumes  of 
the  cube  and  pyramid. 

496.  If  a  pyramid  is  cut  by  a  plane  parallel  to  its  base, 
the  pyramid  cut  off  is  similar  to  the  first,  and  the  two  pyra- 
mids are  to  each  other  as  the  cubes  of  any  two  homologous 
edges.     Prove. 


REGULAR  POLYEDRONS. 
Definition. 

A  regular  poJyedron  is  a  polyedron  whose  faces  are  equal 
regular  polygons  and  whose  dicdral  angles  are  all  equal. 
[See  §456  for  figures.] 

Questions. 

534. 

1.  What  is  the  fewest  number  of  faces  necessary  to 
form  a  polyedron?     Name  the  solid. 

2.  How  many  faces  meet  at  each  vertex  of  the  tetraedon. 

3.  What  is  the  fewest  number  of  faces  required  to  form 
a  convex  polyedral  angle?     What  is  it  called  ? 

4.  Illustrate  how  the  sum  of  the  face  angles  of  any  polye- 
dral angle  compares  with  4  right  angles. 

5.  What  is  the  angle  of  an  equilateral  triangle  ?  How 
many  equilateral  ^ '\S  are  required  to  form  a  solid  angle  ?  What 
is  the  sum  of  the  face  angles  forming  the  solid  angle  ?     Could 


REGULAR  POLYEDRONS.  309 

a  convex  polyedral  angle  be  formed  with  four  equivalent 
As?  Illustrate.  With  five?  Illustrate.  With  six?  Illustrate. 
How  many  regular  solid  figures  may  be  made  using  equi- 
lateral /^s? 

6.  What  is  the  limit  of  the  number  of  squares  that  can 
be  used  in  forming  regular  convex  polyedrons?  Illustrate. 
How  many  convex  polyedrons  can  be  formed  with  squares? 
Name  the  solid. 

7.  How  many  regular  pentagons  may  be  used  to  form  a 
solid  angle  ?  Illustrate.  How  many  convex  polyedrons  can 
be  formed  with  regular  pentagons?     Name  the  solid. 

8.  What  is  the  limit  of  the  number  of  regular  polyedrons 
formed  of  regular  hexagons?     Why? 

9.  What  is  the  limit  of  the  number  of  sides  of  a  regular 
polygon  that  can  be  used  in  forming  regular  polyedrons. 

10.  What  is  the  greatest  number  of  regular  convex 
polyedrons?  Name  those  that  are  bounded  by  triangles ;  by 
squares:  by  pentagons.  These  five  regular  polyedrons  are 
called  the  Platonic  Bodies. 

635. 

The  point  within  a  regular  polyedron  equally  distant  from 
the  sides  is  called  the  ce7iter  of  the  polyedron. 

How  is  the  center  located  from  the  vertices  ?     Prove  it. 

What  is  the  locus  of  all  points  equally  distant  from  a 
given  point  ? 

Do  you  see  how  a  regular  polyedron  may  be  related  to 
the  surface  of  a  sphere  ? 

[See  §  456  for  diagram  of  the  Platonic  Bodies.] 


810  SOLID  GEOMETRY,  BOOK  VII. 

536. 

Proposition  XXVI. 

Given  the  line  A B. 

Problem:      With  an  edge  equal  to  A  B^  can  you 
construct  a  regular  tetraedron  ? 

(1)  How  many  faces  are  required  for  the  solid? 

(2)  Where  is  the  center  of  a  regular  tetraedron  ?     Let 
V  —  A  B  C  be  the  tetraedron  required. 


(3)  Can  you  find  the  center  of  the  base  ABC?     How 
do  you  erect  a  perpendicular  to  a  plane  at  a  given  point  ? 

(4)  In  what  perpendicular  to  the  base  does  the  vertex  of 
the  tetraedron  lie  ?     How  can  you  find  the  required  point  ? 

537. 
Proposition  XXVII. 


Problem:     Construct  a  regular  hexaedron  whose 
edge  equals  a  given  line  ? 


REGULAR  POLYEDRONS. 


311 


538. 

Proposition  XXVIII. 

Problem:     Consttucl  a  regular  octaedron  whose 
edge  equals  a  given  line. 

539. 
Proposition  XXIX. 


Problem:      Construct  a  regular  icosaedr on. 

[Hint.  1.  With  A  B  as  a  side,  construct  the  regular  pen- 
tagon A  B  C  D  E.  Find  the  center  and  erect  a  1.  Where  on 
this  ±  must  a  point  be  taken  to  form  a  regular  pentagonal 
pyramid?     Let  the  point  be  N. 


Fig.  2.  Fig  3. 

2.     With  A  and  B  as  vertices,  form  pentaedral  angles. 
Study   Fig.  2.     How  many  pentaedral   Z  s  in    Fig.  2  ?     The 


812  SOLID  GEOMETRY,  BOOK  VII. 

parts  of  how  many  more  ?  How  many  faces  at  G?  at  H  ?  at  C  ? 
etc?     How  many  faces  used  in  Fig.  2  ?  ] 

To  construct  (2):  Does  N  B=  A  B?  Why?  Does 
N  E=:  N  B? 

In  the  plane  determined  by  N  B  and  N  E  form  a  regular 
pentagon  having  N  a  vertex.  IVt/l  N  B  and  N  E  be  sides  of 
that  p€7itagon  ? 

Where  is  A,  with  reference  to  the  plane  of  N  B  and  N  E  ? 
Where  is  it,  with  reference  to  the  points  B,  N,  and  E?  With 
center  A  and  radius  A  B,  describe  2i  Q  m.  the  plane  of  N  B 
and  N  E.     Will  B  N  and  N  E  be  chords  of  that  circle  ? 

It  is  necessary  to  prove  that  /_  B  N  E  is  the  L  of  a  regular 
pentagon.  Compare  it  with  /_  B  A  E.  In  what  plane  is  Z 
B  A  E  ?     How  large  is  it  ? 

Caji  yon  prove  the  two  As  eqnal f  If  so,  you  can  con- 
struct a  second  pentaedron  whose  vertex  is  A. 

»/^.  Take  two  points,  x  andj'  on  a  N,  making  ;r  A  =  j  N. 
Through  x  pass  a  plane  perpendicular  to  A  N,  cutting  the 
plane  of  A  B  C  D  E  in  2^  ze/,  z  being  in  A  E  and  w  in  A  B. 
Through  jv  pass  a  plane  perpendicular  to  A  N,  cutting  plane 
of  B  N  E  in  ^'  w ,  z  being  in  N  E,  and  w'  being  in  B  N. 
What  Zs  are  we  trying  to  prove  equal?  In  which  /\s  are 
these  Zs?  Can  you  prove  these  A,s=?  Try  to  prov^e  the 
3  sides  of  one  =  to  3  sides  of  the  other  by  proving  — 

(1)  l\zP^x^  i\z'^y. 

(2)  A  A  ;»;  ze/  =  A  N  y  w' . 

(3)  .\  z  X  w  ^=  z  y  w. 

(1)  (a)     Kx=^y,     [?] 

{b)     A  z  Kx=  I  /Nj,      [?] 

{c)     /,  K  X  z  =^  ^  y  z'\    [Prove  each  is  a  right  Z  ]. 

.'.  l\  z  Ps.x=  l\z  ^  y. 

(2)  Use  method  similar  to  that  used  in  (1). 

(3)  /_  z  X  w  =i  A  z  y  w\     [?] 
.'.  A  etc.,  etc. 

:.  /\  z  k  w=  /\  z  ^  w     (?) 


REGULAR  POLYEDRONS. 


818 


and  Z  E  N  B  r^  Z  E  A  B, 

and  Z  E  N  B  is  an  Z  of  a  pentagon, 

and  N  E  and  N  B  are  sides  of  a  regular  pentagon, 

and  A  is  the  vertex  of  a  second  pentaedral  Z  . 

Thus  we  can  construct  a  regular  pentaedral  at  B. 

3.     Construct  a  convex  surface  like  Fig.  2.    Call  it  Fig.  3. 

Show  that  the  convex  suface  of  Fig.  2  can  be  made  to 
combine  with  that  of  Fig.  3  so  as  to  form  the  required 
icosaedron. 

Note. — The  pupil  will  have  his  ideas  cleared  concerning 
the  Platonic  Bodies  by  making  them  out  of  cardboard.  Some 
of  these  forms  of  regular  polyedrons  are  studied  in  chemistry, 
botany,  geology,  and  crystallography. 

540. 

Proposition  XXX.— Eui^er's  Theorem. 

Can  it  be  proven  that  the  number  of  edges  in- 
creased by  two  in  any  polyedron  equals  the  number 
of  vertices  increased  by  the  number  of  faces? 


I,et  A  B  C  D  be  one  face  of  any  polyedron.  Let  E  = 
number  of  edges;  V  =  number  of  vertices;  F  =:  number  of 
faces.     Does  E  +  2  =  V  +  F? 

Sug.  1.  In  face  A  B  C  D  there  are  4  edges  and  4  vertices; 
i.  e.,  E,  =  V. 

Now  add  the  face  B  C  G  F  by  placing  its  edge  B  C  to  the 


314  SOLID  GEOMETRY,  BOOK  VII. 

edge  B  C  of  the  first  face.  How  many  vertices  in  common  ? 
edges?  How  do  the  edges  and  vertices  compare  thus  far?  Is 
this  true  (1)  E  ==  V  +  1  for  the  two  surfaces  joined? 

Sug.  2.     Now  add  face  A  B  F  E. 

How  many  edges  are  in  common  with  the  first  two  faces? 
How  many  vertices  in  common  with  the  first  two  faces? 

Now  the  total  number  of  edges  is  how  many  more  than 
the  total  number  of  vertices? 

Is  the  following  equation  true  for  the  three  faces  joined? 
(2)  E  =  V  +  2  . 

Sug.  8.     Add  the  face  A  D  H  E. 

How  many  edges  are  in  common  with  the  first  three? 

How  many  vertices  in  common  with  the  first  three. 

The  total  number  of  edges  is  how  many  more  than  the 
total  number  of  vertices? 

Can  you  show  that  (3)  E  ==  V  +  3  when  four  faces  are 
joined? 

Sug.  4.  Continuing  the  process  till  every  face  but  one 
has  been  annexed,  what  effect  is  produced  on  the  right  mem- 
ber of  the  equation  as  each  additional  face  is  added?  At  any 
stage  of  the  process  of  adding  faces,  E  ^=  V  -f  a  number  less 
by  one  than  the  number  of  faces.     Study  (1),  (2),  (3). 

Sug.  5.  In  the  incomplete  solid  lacking  one  face  the 
number  of  faces  is  F  —  1. 

.-.  E  =  V  +  (F  -  1)  —  1,  or  E  =  V  +  F  —  2.  (Pupil  will 
finish  this  proof.) 

Remark. — The  pupil  should  test  this  law  with  the  Pla- 
tonic Solids.  Find  out  what  you  can  about  Euler,  the  math- 
ematician. 

541. 

Proposition  XXXI. 

Considering  the  faces  of  a  polyedron  as  separate  polygons, 
how  does  the  number  of  edges  compare  with  the  number  of 
edges  in  the  polyedron  ? 


REGULAR  POLYEDRONS.  315 

Use  the  notation  of  Prop.  XXX.  and  call  the  sum  of  all 
the  angles  of  the  faces  of  the  polyedron  S,  and  let  R  ^  a 
right  angle. 

Find  an  expression  for  S  in  terms  of  R  and  V. 

Sug.  Form  an  exterior  angle  at  the  vertex  of  each 
polygon. 

What  is  the  sum  of  the  interior  and  the  exterior  angles  at 
each  vertex  ? 

Since  the  whole  number  of  edges  of  the  polygons  which 
form  the  face  of  the  polyedron  =  2  E  (first  quarter),  then  the 
sum  of  all  the  interior  and  exterior  angles  of  the  faces  is 
2  R  .  2  E  =  4  R    E. 

What  is  the  sum  of  the  exterior  angles  of  any  polygon  or 
face  ? 

Then  the  sum  of  all  the  exterior  angles  is  F  •  4  R; 

.'.  the  sum  of  all  the  interior  angles  of  the  sum  of  the 
face  angles  is  (I)  4  R  E  —  4  R  -  F  =r  4  R  (E  —  F),  or  S  = 
4  R  (E  —  F). 

But  from  Prop.  XXX.  E  -f  2  -^  V  +  F,  or,  transposing, 
E  —  F  --  V  -  2.  Substituting  this  value  of  E  —  F  in  (1). 
we  have  S  =-  4  R  (V  —  2). 

Write  the  interpretation  of  the  last  equation,  and  call  it 
Prop  XXXI. 


313 


SOLID  GEOMETRY,  BOOK  VIII, 


BOOK    VIII. 

THE  CYI.INDER. 

Definitions. 

542. 

A  cylindrical  surface  is  a  curved  surface  formed  by  a 
moving  straight  line  which  constantly  touches  a  given  curve 
and  is  at  all  times  parallel  to  a  fixed  straight  line  not  in  the 
plane.  The  moving  straight  line  is  called  i\iQ.ge?ieratrix.  The 
given  curved  line  is  called  the  directrix. 


543. 


Any  straight   line  in  the  cylindrical  surface  is  called  an 
element  of  the  surface,  sls  x  y  above. 

There  may  be  any  number  of  cylindrical  surfaces. 


THE  CYLINDER. 

544. 


317 


A  cylinder  is  a  solid  bounded  by  a  cylindrical  surface  and 
two  parallel  planes.  The  plane  surfaces  are  called  the  bases 
and  the  curved  surface  th^  lateral  or  convex  surface  of  the 
cylinder. 

(How  do  the  elements  compare  in  length?) 


545. 

A  line  joining  the  centers  of  the  bases  is  calltd  the  axis 
of  the  cylinder. 

546. 

A  fight  section  of  a  cylinder  is  the  intersection  of  the  cyl- 
inder and  a  plane  perpendicular  to  an  element. 

547. 

The  altitude  of  a  cylinder  is  the  perpendicular  distance 
between  its  bases. 

548. 

An    oblique  cylinder   is   one   whose   bases   form  oblique 
angles  with  the  elements. 


318  SOLID  GEOMETRY,  BOOK  VIII. 

549. 

A  right  cylinder  is  one  whose  elements  are  perpendicular 
to  its  bases. 

550. 

A  circular  cylinder  is  one  whose  bases  are  circles. 

651. 

In  how  many  ways  may  a  right  circular  cylinder  be 
generated  ? 

552. 

A  right  circular  cylinder  generated  by  the  revolution  of  a 
rectangle  about  one  of  its  sides  as  an  axis  is  called  a  cylinder 
of  revolution. 

553. 

Similar  cylinders  of  revolution  are  generated  by  the  revo- 
lution of  similar  rectangles  about  homologous  sides. 

554. 

An  axial  section  is  formed  by  passing  a  plane  through  the 
axis. 

555. 

A  plane  is  tangent  to  a  cylinder  when  it  passes  through 
an  element  of  the  cylinder,  but  does  not  cut  the  surface. 

556. 

A  tangent  line  to  a  cylinder  is  a  line  which  touches  the 
cylindrical  surface  at  a  point,  but  does  not  intersect  the  surface. 

557. 

Remark:  The  generatrix  is  supposed  to  be  indefinite  in 
extent;  hence  the  surface  generated  is  also  of  indefinite  extent. 


THE  CYLINDER.  319 

558. 

Proposition  I. 

Can  you  show  that  any  section  of  a  cylinder 
made  by  passing  a  plane  through  an  element  is  a 
parallelogram  ? 


Let  the  plane  A  C  contain  an  element  A  B.  Is  A  B  C  D 
a  parallelogram? 

Siig.  Is  point  D  common  to  the  plane  and  the  surface  of 
the  cylinder?  Draw  a  line  through  D  ||  to  A  B.  Where  will 
it  lie  ? 

Pupil  complete  proof  and  write  the  proposition,  calling  it 
Prop.  I. 

559. 

Cor,     Every   section  of  a  right  cylinder   made 

by  passing  a    plane   through  an  element  is  a   . 

Proof. 


320  SOLID  GEOMETRY,  BOOK  VIII. 

560. 

Proposition  II. 

Can   you   show  that  the  bases   of  a  cylinder  are 
equal? 


Sug.  1.  Suppose  A  B'  to  be  any  cylinder  and  F,  G,  any 
two  points  in  the  perimeter  of  the  upper  base.  Pass  the  plane 
F  D  containing  the  line  F  G  and  the  element  F  C.  Compare 
F  G  and  C  D 

Sug.  2.  Take  H  as  any  other  point  in  the  perimeter  of 
the  upper  base  and  H  E  an  element  through  H.  Pass  planes 
through  H  E,  F  C  and  H  E,  G  D.  Compare  F  H,  C  E  and 
G  H,  D  E. 

Sug.  3.  Compare  As.  Superpose  the  upper  base  on 
the  lower  base  so  F  G  will  fall  on  C  D  and  A  E  G  H  wil 
fall  on  A  C  D  E.   What  follows? 

Make  the  generalization.     Write  it  and  call  it  Prop.  II. 


CYLINDERS. 


321 


561. 

Cor.  L  In  the  cylinder  A  B  let  C  D  and  E  F 
be  1  sections  made  by  I  planes  cutting  all  the 
elements.      Compare  the  sections. 


c     , 


l^^lg.  1- 


Fig   2. 


562. 


Cor.  II.  Compare  the  section  of  a  cylinder 
made  by  a  plane  ||  to  the  base  with  the  base. 

563. 

Cor,  III  Can  3^ou  show  that  the  axis  of  a 
circular  cylinder  passes  through  the  centers  of  all 
the  sections  ||  to  the  base  ? 

Sug.  Draw  any  two  diameters  of  one  of  the  bases,  as 
D  E,  F  G.  Through  these  diameters  and  elements  of  the  cyl- 
21 — 


322 


SOLID  GEOMETRY.  BOOK  VIII. 


inder    pass  planes  cutting  the  upper  base  in  H  I  and  J  K. 
Are  H  I  and  J  K  diameters? 
Finish  proof. 

564. 

Cor.  IV.  Suppose  AC  to  be  the  axis  of  a  cir- 
cular cylinder.  Can  you  show  that  it  is  equal  and 
parallel  to  the    elements  of  the  cylinder? 

565. 

A  cylinder  is  said  to  be  inscribed  in  a  prism  when  each 
lateral  face  is  tangent  to  the  cylinder  and  the  bases  of  the 
prism  circumscribe  the  bases  of  the  cylinder.  The  prism  is 
also  said  to  be  circurnscribed  about  the  cylinder. 


566. 

A  cylinder  is  circumscribed  about  a  prism  when  each  edge 
of  the  prism  is  an  element  of  the  cylinder  and  the  bases  of  the 
prism  are  inscribed  in  the  bases  of  the  cylinder.  The  prism 
is  said  to  be  inscribed  in  the  cylinder. 


CYLINDERS.  323 

567. 

Proposition  III. 
Let  H  C  be  any  cylinder  and  I  L  a  right  section. 
How  do  we  compute  the  convex  surface  ? 


Sug.  1.  How  find  the  lateral  surface  of  any  prism?  If 
the  lateral  faces  of  the  inscribed  prism  H  C  are  indefinitely 
increased,  to  what  solid  does  the  prism  approach  as  a  limit? 
How,  then,  does  the  convex  surface  of  any  cylinder  compare 
with  the  rectangle  formed  by  any  element  and  the  perimeter 
of  a  right  section  ? 

Siig.  2.  Let  the  number  of  sides  of  H  C  be  increased  by 
bisecting  the  arcs  subtended  by  the  sides,  and  joining  the 
points  of  bisection. 

Sug  3.  What  is  the  limit  of  the  lateral  area  of  the  prism 
as  the  sides  are  indefinitely  increased  ?  What  does  the  per- 
imeter of  the  polygon  approach  as  a  limit? 

Sug.  4.  As  the  lateral  area  increases,  what  is  the  ex- 
pression for  its  surface  ? 


324  SOLID  GEOMETRY,  BOOK  VIlI. 

Siig.  5.  Call  convex  surface  of  cylinder  S  and  circumfer- 
ence of  I  L,  P,  and  let  S'  =  surface  of  the  prism  H  C  and 
P'  =  the  perimeter  of  rt.  section  I  ly. 

(1)  S'  is  equivalent  to  G  D  F  Why?  But  S' =  ? 
G  D     P  =? 

Is  (1)  always  true  for  any  number  of  sides  ? 

Are  both  sides  of  (1)  variables? 

Pupil  will  finish  the  proof  and  write  the  general  state- 
ment, calling  it  Prop.  III. 

568. 

Cor.  How  find  the  lateral  surface  of  a  cylinder 
of  revolution  ? 

Two  cylinders  are  said  to  be  similar  when  the  ratio  of  the 
axis  to  the  radius  of  the  base  of  the  one  equals  the  ratio  of 
the  axis  to  the  radius  of  the  base  of  the  other. 

ExKRCrsK. 

497.  If  S  =  lateral  surface  of  a  cylinder,  k  =  altitude, 
r  =  radius  of  the  base,  and  A  =  total  or  entire  surface,  find 
an  expression  for  A  in  terms  of  //,  r,  and  tt. 

569. 

Proposition  IV. 

What  are  similar  rectangles? 

What  are  similar  cylinders?     Illustrate. 

Compare  the  lateral  areas  of  two  similar  cylinders  whose 
altitudes  are  6  inches  and  2  inches  respectively,  and  whose 
radii  are  3  inches  and  1  inch  respectively. 

How  does  their  ratio  compare  with  the  ratio  of  the  squares 
of  the  radii  ?  with  the  ratio  of  the  squares  of  the  altitudes? 

Can  you  show  that  the  lateral  areas  of  similar 
cylinders    of  revolution    are    to  each   other    as  the 


CYLINDERS. 


325 


squares   of  their   radii,    or  as   the    squares  of  their 
altitudes  ? 

Sug.  1.  Let  S,  k,  r  and  S'  fi ,  r  represent  the  lateral  sur- 
face, altitude,  and  radius  of  each  of  the  similar  cylinders  (1) 
and  (2). 


Fig.  (1.) 


Fig.  (2. 


r        h 

Sug.  2.     Can  you  prove  that— ,  =  t>? 

Convex  surface  of  a  cylinder  of  revolution  S  ^  2  tt  r  ^. 
Why?     [§359.] 

8        2  IT  r     h        r    h         r  .  h  . 


'    •S'~2^//fe'~/  -// 


h! 


What  does  —  =  ? 
r 


Write  the  generalization,  and  call  it  Prop.  IV. 


826 


SOLID  GEOMETRY,  BOOK  VIII. 


570. 

Cor.     By  Ex.  497  can  you  show  that  the  entire 
surface  of  Fig.  (1)  is  to  entire  surface  of  Fig.  (2)  as 


^/2  or  as  ^,j 


571. 
Proposition  V. 

Can  you  prove  that  the  volume  of  a7iy  cylinder 
equals  the  product  of  its  base  by  its  altitude  ? 

Sug.  1.     How  do  you  find  the  volume  of  any  prism  ? 

If  the  lateral  faces  of  the  prism  be  indefinitely  increased, 
to  what  does  the  prism  approach  as  its  limit  ? 


Sug,  2.  Let  G  C  be  any  cylinder  and  inscribe  a  prism 
within  the  cylinder.  Now  let  the  number  of  sides  increase 
indefinitely. 

Sug.  3.     For  convenience    call  volume  of  the  cylinder 


CYLINDERS.  327 

V,  altitude  k,   base   b,    and  volume   of  prism   v',  altitude  h' , 
base  b'. 

z;'  =  ?     For  any  number  of  sides  .'' 

v'  =  ^     b'  =  } 

Draw  and  state  conclusion,  and  call  it  Prop.  V. 

572. 

Cor.  I.  Call  r  the  radius  of  a  cylinder  of  revolu- 
tion, and  show  that  v  ='  "^  r^^  h. 

573. 

Cor.  II.  Can  you  show  that  the  volumes  of  any 
two  similar  cylinders  of  revolution  are  to  each  other 
as  the  cubes  of  their  radii,  or  as  the  cubes  of  their 
altitudes  .^ 

Sug.  1.    (1)7  =  ^,.  Why? 
v' =  iTr'*.h!',     .-.-,  =  ? 

V 

Complete  proof. 

574. 

Cor.  III.  Can  you  show  that  similar  cylinders 
of  revolution  are  to  each  other  as  the  cubes  of  any 
like  dimensions  ? 


828  SOLID  GEOMETRY,  BOOK  VIII. 

THE  CONE. 

Definitions. 

575. 

A  conical  surface  is  the  surface  generated  by  a  moving 
straight  line  called  the  generatrix,  passing  through  a  fixed 
point  and  constantly  touching  a  fixed  curve. 

576. 

The  fixed  point  is  called  the  vertex,  and  the  fixed  curve 
is  called  the  directrix.  Let  the  figure  represent  a  conical 
surface. 

577. 


The  generatrix  in^any  position,  as  A  A'  C  C,  is  called  an 
element  of  the  surface. 

578. 

When  the  generatrix  is  of  indefinite  length,  the  surface  is 
composed  of  two  portions,  one  above  the  vertex,  the  other 
below  the  vertex.  These  parts  are  called  the  upper  and  lower 
nappesy  respectively. 


CONES.  329 

579. 

A  cone  is  a  solid  bounded  by  a  conical  surface  and  a  plane. 
The  plane  is  called  the  base  of  the  cone,  the  conical  surface  is 
called  the  lateral^  or  convex  surface. 

580. 

The  altitude  of  a  cone  is  the  perpendicular  distance  from 
the  vertex  to  the  base,  or  the  base  produced.  The  axis  of  a 
cone  is  the  line  joining  its  vertex  to  the  center  of  its  base. 

581. 

A  circular  cone  is  one  whose  base  is  a  circle. 

582. 

If  the  axis  is  perpendicular  to  the  base,  the  cone  is  called 
a  right  cone. 

583. 

If  the  axis  is  oblique  to  the  base,  it  is  called  an  oblique 
cone. 

584. 

A  right  circular  cone  is  called  a  cone  of  revolution^  since 
it  may  be  generated  by  revolving  a  right  triangle  about  one 
of  its  legs  as  an  axis. 

What  does  the  hypotenuse  generate?  the  leg  not  used 
for  the  axis?     What  is  the  hypotenuse  in  any  position? 

The  element  of  a  right  cone  is  the  slant  height  of  the  cone. 

All  the  elements  of  a  coiie  of  revolution  are  equal.     Why  ? 

Has  an  oblique  cone  a  slant  height  ? 

585. 

Similar  cones  of  revolution  are  cones  generated  b}^  similar 
right  triangles  revolving  about  homologous  legs. 


330 


SOLID  GEOMETRY,  BOOK  VIII. 
586. 


A  tangent  line  to  a  cone  is  a  line,  not  an  element,  which 
touches  the  cone,  but  does  not  cut  it,  no  matter  how  far  it  is 
produced. 

587. 

A  tangent  plane  contains  an  element  of  the  cone,  but  does 
not,  when  produced,  cut  the  surface. 

The  elevient  of  contact  is  the  element  contained  by  the 
tangent  plane. 


588. 


A  pyramid  is  inscribed  in  a  cone  when  its  lateral  edges  are 
elements  of  the  cone  and  its  base  is  inscribed  in  the  base  of 
the  cone. 


CONES. 


331 


589. 


A  pyramid  is  circumscribed  about  a  cone  when  its  base 
circumscribes  the  base  of  the  cone  and  its  vertex  coincides 
with  the  vertex  of  the  cone. 


590. 


The  frustum  of  a  cone  is  that  part  of  the  cone  included 
between  the  base  and  a  section  parallel  to  the  base.  The  base 
of  the  cone  is  called  the  lower  base  of  the  frustum,  and  the 
parallel  section  is  called  the  upper  base  oj  the  frustum. 


832  SOLID  GEOMETRY,  BOOK  VIII. 

591. 

The  altitude  of  the  frustum  is  the  perpendicular  distance 
between  the  bases. 

592. 

The  slant  height  of  a  frusturd  of  a  cone  of  revolution  is  that 
part  of  the  slant  height  of  the  cone  which  is  included  between 
the  bases. 

593. 

Proposition  VI. 


Can  you  prove  what  section  is  formed  by  passing 
a  plane  through  any  part  of  the  cone  and  its  vertex? 

What  is  V  C  D?     Prove  it. 

Write  the  generalization,  and   call  it  Prop.  VI. 

What  is  the  section  when  the  cone  is  a  cone  of 
revolution  ? 


CONES. 

594. 

Proposition  VII. 


333 


Can  you  show  what  any  section  of  a  circular 
cone  is  when  formed  by  a  plane  parallel  to  the  base? 

Sug.  1.  Suppose  V  O  the  axis  of  the  cone  and  O'  the 
point  where  the  axis  pierces  the  section. 

Sug.  2.  Through  axis  V  O  pass  any  plane  intersecting 
the  base  in  A  O  and  the  section  in  A'  O'.  In  the  same  man- 
ner through  V  O  pass  any  other  plane  intersecting  the  base 
in  O  C  and  the  section  in  O'  C. 

Sug.  3.     Compare  O'  A'  and  O  A,  also  O'  C  and  O  C. 


Proof: 

0) 

(2) 

(3) 

S7ig.  4. 


V  O'  :  V  O  ::  O'  A'  ::  O  A.     [  ?  ] 

VO'  :  VO::  O'C  :  O  C.     Why? 

O' A':OA::0'C':  O  C.     Why? 

What  is  the  base  of  the  cone  .'* 
Now  O  A  =  O  C.     Why  t 
.'.  O'A'^  O'C       Why? 

But  A'  and  C  are  any  two  points  in  the  perimeter  of  the 
section;  therefore — 


334 


SOLID  GEOMETRY,  BOOK  VIII. 


595. 

Cor,     How  does  the  axis  of  a  circular  cone  pierce 
sections  parallel  to  the  base?     Proof. 

596. 

Proposition  VII. 

Can  you  find  an  expression  for  the  convex  sur- 
face of  a  cone  of  revolution? 


Sug.  Inscribe  a  regular  pyramid  within  the  cone 
V— A  BCD. 

How  find  the  lateral  surface  of  the  pyramid  ? 
Complete  the  demonstration. 
See  method  in  Prop.  III. 

597. 

Cor.  Let  S=  convcK  surface  of  a  right  cone,  rz=z  radius 
of  the  base,  and  s  =  slant  height.  Show  that  S--^{27rrs) 
=z  TT  r  •  s.  If  A  =  total  area,  show  that  A  =7r  r{s  ^  r).  These 
are  convenient  formulae  in  arithmetic. 

Exercises. 
498.     The  lateral  areas  of  two  similar  cones  of  revolution 
are  to  each  other  as  the  squares  of  their  slant  height,  or  the 


CONES.  335 

squares  of  their  altitudes,  or  the  squares  of  the  radii  of  their 
bases. 

[Hint.— See  Prop.  IV.] 

499.  The  altitudes  of  two  similar  cones  of  revolution  are 
8  inches  and  4  inches,  slant  heights  10  inches  and  5  inches,  and 
the  radii  of  their  bases  6  inches  and  8  inches.  Compare  total 
areas  with  the  ratio  of  the  squares  of  their  altitudes. 

500.  If  a  cone  is  circumscribed  about  a  pyramid,  prove 
that  each  lateral  edge  of  the  pyramid  is  an  element  of  the 
cone. 

598. 
Proposition  VIII. 

How  do  you  find  the  lateftil  surface  of  the  frustum  of  a 
right  pyramid? 

As  the  number  of  sides  of  the  frustum  of  the  pyramid  is 
indefinitely  increased,  to  what  solid  does  the  frustum  of  the 
pyramid  approach  ? 

Can  you  show  that  the  convex  surface  of  a  frus- 
tum of  a  cone  of  revolution  equals  one-half  the  prod- 
uct of  its  slant  height  by  the  sum  of  the  circumfer- 
ences of  its  bases  ? 

Sag.     See  method  of  Prop.  III. 

Write  the  proposition.  Can  you  make  an  arithmetical 
formula  for  this  proposition  ? 


336  SOLID  GEOMETRY,  BOOK  VIII. 

599. 

Proposition  IX. 

How  do  you  compute  the  volume  of  a  pyramid  ?  To  what 
doCvS  the  pyramid  approach  as  its  limit  if  the  sides  be  increased 
indefinitely? 

Can  you  prove  that  the  volume  of  a  cone  equals 
one-third  of  the  product  of  its  base  by  its  altitude  ? 


Sug.  Inscribe  a  pyramid  within  the  cone.  Pupil  com- 
plete demonstration. 

600. 

Cor.  Show  that  if  z/  =  volume  of  circular  cone, 
the  formula  z/=  \  -^  v'^  h\s>  true  if  ^  =  radius  of  base, 
and  h  =  altitude  of  the  cone. 


CONES. 


337 


601. 

Proposition  X. 

What  are  similar  cones  of  revolution  ? 

Compute  the  volumes  of  two  cones  whose  altitudes  are 
6  inches  and  3  inches  and  whose  diameters  are  8  inches  and 
4  inches. 

Compare  their  volumes  with  the  cubes  of  the  altitudes ; 
with  the  cubes  of  the  radii  of  their  bases;  with  the  cubes  of 
the  diameters  of  their  bases. 


Given  (1)  and  (2),   two  similar  cones  of  revolution  whh 
/i,  Ji  the  respective  altitudes  and  r,  /  the  respective  radii. 

Can  you  show  that  the  volumes  of  these  cones 
are  to  each  other  as  the  cubes  of  their  altitudes,  or 
as  the  cubes  of  the  radii  of  their  bases,  or  as  the 
cubes  of  the  diameters  of  their  bases? 

Sug.     Let  V  =  volume  of  (I)  and  v  =  volume  of  (2). 

V  ~  '        h' 


Vr=? 


—  ? 


i=. 


Write  the  conclusion,  and  call  it  Prop.  X. 


338  SOLID  GEOMETRY,  BOOK  VIII. 

602. 

Proposition  XI. 

How  do  you  find  the  volume  of  the  frustum  of  any 
pyramid  ? 

As  the  number  of  sides  is  indefinitely  increased,  to  what 
does  the  frustum  of  the  pyramid  approach  as  a  limit? 


I^et  A  E  C  D  I —  E'  be  an  inscribed  frustum  of  a  pyra- 
mid. Its  volume  =-  V'.  Call  h  the  altitude,  B'  the  area  of  the 
lower  base,  b'  the  area  of  the  upper  base,  and  \/  B'  b'  the  area 
of  the  mean  base.    [P.  P.]    Call  V  the  volume  of  the  cone. 

SugA.     §519,  V  =  1  /^(B'4-^'+   V^B^T^). 

Increase  the  lateral  faces  of  the  frustum  of  the  pyramid 
indefinitely.  To  what  does  B'  ^i^  ?  <^' -i- ?  V'  =l=  ?  Is  the  above 
equation  always  true  for  any  number  of  sides  ?  Apply  law  of 
limits  and  complete  the  demonstration. 

Exercise. 

501.  If  r  and  r'  denote  the  radii  of  the  base,  can  you 
prove  that  v  =  \ir  h{r^  -\-  {r'f    +  r  /). 


EXERCISES.  339 

KxKRCISES. 

502.  If  B  r=  ba^e,  b  =  upper  base,  C^  circumference 
of  base,  c  =  circumference  of  upper  base,  c  =  circumference 
of  mid-section,  r  =  radius  of  base,  /  =  radius  of  upper  base, 
k  =  altitude,  d  =  diameter,  s  =  lateral  area,  S  =  total  area, 
and  V  =  volume. 

Write  formulas  for  a  cylinder  of  revolution,  a  cone  of  rev- 
olution, and  the  frustum  of  a  cone  of  revolution. 

503.  Find  s,  S,  and  v  by  the  formula  above  when  d  =  8 
dm.  and  ^  =  10  dm. 

504.  The  dimensions  of  the  frustum  of  a  cone  of  revolu- 
tion are  r  =  6,  /  =  4,  and  //  =  9.     Find  s,  S,  and  v. 

505.  How  many  square  yards  of  tin  will  be  required  to 
cover  a  conical  tower  whose  base  is  12  feet  in  diameter  and 
whose  altitude  is  20  feet? 

508.  A  ship's  mast  is  40  feet  long,  12  inches  in  diameter 
at  the  lower  base,  and  5  inches  at  the  upper  base.  Find  its 
volume. 

507.  A  cylindrical  cis'^ern  is  5  m.  deep  and  2J  m.  in  diam- 
eter. If  2  HI.  flow  into  it  per  minute,  how  long  will  the  cistern 
be  in  filling? 

508.  Can  you  prove  that  the  lateral  surface  of  a  pyra- 
mid circumscribed  about  a  cone  is  tanget  to  the  cone? 

509.  In  a  cylinder  of  revolution  d=/i,  can  you  show  that 

510.  Two  right  circular  cylinders  have  the  diameter  and 
the  altitude  of  each  equal.  Ifthe  volumeof  oneis-rrjof  that  of 
the  other,  what  is  the  relation  of  their  altitudes  ? 


340  SOLID  GEOMETRY,  BOOK  VIII. 

THE  SPHERE. 
Definitions. 

603. 

A  sphere  is  a  solid  bounded  by  a  surface  all  points  of  which 
are  equally  distant  from  a  point  within  called  the  center. 

604. 

The  radius  of  a  sphere  is  the  distance  from  the  center  to 
any  point  on  the  surface.  ^ 

605. 

The  diameter  of  a  sphere  is  any  straight  line  passing 
through  the  center  terminated  by  the  surface  of  the  sphere. 
It  follows  that  all  radii  are  equal,  that  any  diameter  is  twice 
the  radius,  and  that  all  diameters  are  equal. 

606. 

A  line  is  tangent  to  a  sphere  when  it  has  only  one  point 
in  common  with  the  sphere. 

A  plane  is  tangent  to  a  sphere  when  it  has  only  one  point 
in  common  with  the  sphere. 

Two  spheres  are  tangent  when  they  have  only  one  point 
in  common. 

607. 

Two  spheres  are  concentric  when  they  have  a  common 
center. 

608. 

A  polyedron  is  said  to  be  inscribed  in  a  sphere  when  its 
vertices  lie  in  the  surface  of  the  sphere.  The  sphere  may  be 
said  to  be  circumscribed  about  the  polyedron. 

A  polyedron  is  circumscribed  about  a  sphere  when  its  faces 
are  tangent  to  the  surface  of  the  sphere.  In  this  case  the 
sphere  is  inscribed  within  the  polyedron. 


THE   SPHERE.  341 

609. 

Proposition  I. 


How  are  points  on  the  surface  of  a  sphere  located  with 
regard  to  the  center  ? 

Let  K  C  F  represent  a  sphere  and  E,  D,  F,  A  be  points  in 
the  intersection  mads  by  cutting  the  sphere  with  a  plane. 

What  is  the  section  E  A  F  D? 

Sug.  1.  Let  A  and  D  be  any  two  points  in  the  intersec- 
tion, and  join  them  with  the  center  of  the  sphere. 

Erect  a  _L  to  the  plane  from  the  center,  C,  meeting  the 
plane  at  B. 

Sug.  2      Compare  As  A  C  B  and  D  C  B,  A  B  and  B  D. 

Draw  conclusion  and  write  it,  calling  it  Prop.  I. 

610. 

Cor.  I.  Can  you  show  how  the  line  drawn  from  the  center 
of  the  sphere  to  the  center  of  a  circle  of  the  sphere  is  related 
to  the  plane  of  the  circle  ? 

611. 

Cor.  II.  Two  circles  are  equally  distant  from  the  center. 
Compare  them. 

612. 

Cor.  III.  Of  two  circles  unequally  distant  from  the  center 
of  a  sphere,  prove  which  is  ihe  larger. 

613. 

Cor.  IV.  Compare  circles  which  contain  the  center  of 
the  sphere. 


342  SOLID  GEOMETRY,  BOOK  VIII. 

614. 

A  great  circle  of  a  sphere  is  one  that  contains  the  center 
of  the  sphere. 

A  small  circle  of  a  sphere  is  one  that  does  not  contain  the 
the  center  of  the  sphere. 

615. 

The  poles  of  a  circle  of  a  sphere  are  the  extremities  of  the 
diameter  which  is  perpendicular  to  the  plane  of  the  circle. 
This  diameter  is  often  called  the  axis  of  the  sphere. 

616. 

What  section  is  formed  if  a  plane  passes  through  the 
center  of  a  sphere  ?     Prove. 

617. 

Proposition  II. 


B 

Let  A  B  C  D  be  any  sphere  cut  by  the  two  intersecting 
great  circles  A  K  C  F  and  B  E  D  F.  E  F  is  the  line  of  inter- 
section of  the  two  circles. 

Compare  the  parts  of  each  great  circle? 


THE  SPHERE.  843 

Sug.  I.  Where  is  the  center  of  the  sphere  with  regard  to 
the  centers  of  the  two  circles  ? 

Sug.  2.  Does  the  line  of  intersection  pass  through  the 
center  of  the  sphere?  Through  the  center  of  the  circles?  What 
does  the  line  E  F  do  to  each  circle? 

,Write  conclusion,  and  call  it  Prop.  II. 

618. 

Proposition  III. 


c 

How  many  points  determine  a  plane? 
How  does  a  plane  cut  a  sphere  ? 

How  many  poiuts  on  the  surface  of  a  sphere  de- 
termine a  circle  ? 

Sug.     Can  you  pass  a  plane  through  points  A,  B,  C  ? 

619. 

Cor.     Is  there  any  circle  which  may  be  deter- 
mined by  two  points  on  the  surface  ? 

Exercise. 

511.     Two  great  circles  have  their  planes  perpendicular 
to  each  other.     How  are  they  related  to  each  other's  poles  ? 


344 


SOLID  GEOMETHY,  BOOK  Vlll. 
620. 


When  the  distance  between  two  points  on  the  surface  of 
a  sphere  is  spoken  of,  the  shorter  of  the  two  arcs  is  meant. 


Thus,  in  the  great  circle  D  E  B  F  the  distance  between 
A  and  B  is  the  shorter  arc. 

621. 

Proposition  IV. 

Let  S  represent  any  sphere,  P  and  P',  the  poles  of  the 
circle  A  B  C  D. 


Can  you  show  that  all  points  of  the  circumfer- 
ence of  A  B  C  D  are   equally  distant  from  P  and  P7 


THE  SPHERE.  345 

Sug.  1.  Talse  any  two  points,  as  A  and  B,  in  the  circum- 
ference of  A  B  C  D,  and  pass  the  arcs  of  great  circles  P  A  and 
P  B.  Let  the  axis  P  P',  pierce  the  plane  A  B  C  D  at  O  and 
draw  O  A,  O  B,  O  C,  O  D. 

Sug.  2,     Compare  O  A,  O  B;  P  A,  P  B;  P'  A,  P'  B. 

Draw  conclusion  and  write  it.    Call  this  theorem  Prop.  IV. 

622. 

The  po/ar  distance  of  a  circle  of  a  sphere  is  the  distance 
from  the  nearer  of  its  poles  to  the  circumference  of  the  circle. 
Thus  in  §  621  P  A  and  P  B  are  polar  distances. 

623. 

The  polar  disiafice  of  a  great  circle  is  a  quurdrant.  Show 
this  from  the  figure. 


Let  S  be  any  sphere,  and  P  be  at  a  quadrant's  distance 
from  two  points,  A  and  B,  on  a  great  circle,  A  B  C  D. 


846 


SOLID  GEOMETRY,  BOOK  VIII. 


To  show  that  P  is  the  pole  of  the  arc  A  B. 

Sug.     Join  the  center  of  the  sphere,  O,  with  A,  B,  P. 

Compare  Z  s  P  O  A  and  P  O  B. 

How  is  P  O  related  to  the  plane  A  B  C  D?  What  is  P  of 
arc  A  B. 

Write  a  general  statement,  and  call  it  Prop.  V. 

Remark:  Discuss  the  question  when  the  two  points  are 
at  the  extremity  of  a  diameter,  as  B  and  D. 

625. 

Scholium:  An  arc  of  a  great  or  small  circle  may  be 
drawn  between  two  given  points  on  the  surface  of  sphere  by 
placing  one  foot  of  the  compasses  at  the  pole. 

The  distance  from  the  point  to  the  pole  must  be  known. 

626. 

Proposition  VI. 


Given  the  plane  M  N  perpendicular  to  the  radius  of  the 
sphere,  S,  at  its  extremity,  A. 


THE  SPHERE.  347 

Can  you  show  how   the    plane  is  related  to  the 
sphere? 

Siig.     Compare  C  A  and  C  B,  B  being  any  point  in  the 
plane  except  A. 

Complete  the  proof.     Write  Prop.  VI. 


627. 

Cor.  I.  Write  and  prove  the  converse  of  Prop. 
VI. 

628. 

G?/-.  //.  Prove  that  a  straight  line  tangent  to 
any  circle  of  a  sphere  lies  in  the  plane  tangent  to 
the  sphere  at  the  point  of  contact. 


629. 

Cor,  III.  Show  that  any  straight  line  in  a  tan- 
gent plane  at  the  point  of  contact  is  tangent  to 
the  sphere. 

630. 

Cor.  IV.  Prove  that  any  two  straight  lines 
tangent  to  a  sphere  at  the  same  point  determine  the 
tangent  plane. 


348  SOLID  GEOMETRY,  BOOK  VIII. 

631. 

Proposition  VII. 


Given  the  sphere  s.  Choose  any  two  points  as,  A,  B,  less 
than  a  circumference  apart  on  a  great  circle  and  with  the  cen- 
ter pass  a  plane.  What  kind  of  a  section  is  formed  ?  What 
kind  of  an  arc  is  A  B? 

Draw  any  other  line  on  the  surface  of  the  sphere 
joining  these  two  points  and  compare  its  length 
with  A  B. 

Sug.  1.  Let  this  line  between  A  and  B  be  A  E  D  B. 
Take  any  point  P  on  A  E  and  D  B  pass  arcs  of  great  circles, 
through  A  and  P,  B  and  P.  Draw  O  A,  O  B,  O  P.  Pass 
planes  through  these  lines. 

Sug.  2.  What  kind  of  an  angle  is  O  —  A  P  B  i*  Compare 
the  face  angle  A  O  B  with  the  sum  of  the  face  angles  A  O  P 
and  BOP.  Compare  the  arc  A  B  with  the  sum  of  the  arcs 
A  P  and  B  P. 

Sug.  3.  Take  any  other  point  in  A  E  P  D  B,  as  C,  and 
connect  it  with  arcs  of  great  circles  with  A  and  P.  Connect 
D  with  P  and  B  in  the  same  manner.  Compare  the  sum  of 
the  great  circle  arcs  A  C,  P  C,  P  D,  D  B  with  the  sum  of  great 
circle  arcs  A  P  and  P  B.     Also  with  A  B. 


THE   SPHERE. 


349 


Sug.  4.  Suppose  we  continue  to  take  points  on  the  line 
A  E  P  D  B  and  proceed  as  above.  Do  you  see  that  the  sum 
of  the  arcs  of  the  great  circles  is  a  variable  ?  What  does  this 
sum  =  ? 

Sug.  5.  As  the  number  of  points  on  A  E  P  D  B  is  in- 
creased, how  does  the  sum  of  the  arcs  compare  with  the  sum 
just  preceding? 

Siig.  6.     How  does  A  E  P  B  D  compare  with  the  arc  A  B? 

Write  the  conclusion,  and  call  it  Prop.  VII. 


Problem, 
sphere  ? 


632. 

Proposition  VI II. 

Can  you  find  the  radius  of  a  physical 


Given'.     Any  sphere,  S. 


Sug.  1.     Take  any  point,  P,  on  the  surface  as  a  pole,  and 
'yvith  any  opening  of  the  compasses  describe  a  circle. 


350  SOLID  GEOMETRY,  BOOK  VIII. 

Take  any  three  points  on  the  circumference,  as  A,  B,  C, 
and  measure  the  distances  A  B,  B  C,  0  A. 

Sug.  2.  Construct  a  triangle,  A'  B'  C,  with  sides  A  B, 
B  C,  C  A,  and  circumscribe  a  circle  about  it.  Call  the  ra- 
dius O'  B'. 

WithO'  B'  as  one  side  of  a  right  triangle  and  P  B  the  hy- 
potenuse, construct  the  triangle  P'  B'  E. 

If  O  is  the  intersection  of  the  diameter  of  the  sphere  and 
the  diameter  of  the  circle  ABC,  compare  As  P  O  B  and 
P'  B'  E. 

Su^.  3.  At  B'  draw  B'  Q  L  P'  B'  and  compare  A  P'  B'  Q 
with  P  B  P'. 

Can  you  measure  P'  Q  ?  Can  you  find  the  radius  of  the 
sphere  ? 

633. 

Cor.  /.  Can  you  find  the  chord  of  a  quadrant 
of  a  sphere  when  the  radius  of  the  sphere  is  known  .^ 

634. 

Cor.  IL  From  Cor.  I.  can  you  describe  a  circum- 
ference of  a  great  circle  through  any  two  points  on 
the  surface  of  a  sphere  whose  radius  =  r? 


Sug.     Find  the  chord  of  a  quadrant  from  Cor.  I.     Take 
any  two  points,  A,  B,  on  the  sphere. 
Pupil  demonstrate. 


THE   SPHERE.  351 

635. 

PropOvSition  IX. 


B 

Bisect  three  diedral  angles  at  the  base  of  a  triangular 
pyramid  with  planes. 

How  does  any  point  in  either  of  the  bisecting  planes  re- 
late to  the  sides  of  the  diedral  angle  bisected  ?  How  will  the 
three  bisecting  planes  meet? 

Given:     Any  tetraedron,  v  —  A  B  D. 

Can  you  inscribe  a  sphere  within  it? 

Write  the  theorem,  and  call  it  Prop.  IX. 

636. 

Cor.  Can  you  prove  how  the  six  bisecting 
planes  of  the  diedral  angles  of  a  tetraedron  meet  ? 

637. 

Proposition  X. 


Imagine  part  of  a  hollow  glass  sphere  to  receive  in  part 
another  smaller  sphere. 


352  SOLID  GEOMETRY,  BOOK  VIII. 

Can  you  picture  the  section  made  by  the  surfaces  coming 
in  contact? 

What  is  the  section  of  the  intersection  of  the 
the  surfaces  of  two  spheres  ?  How  is  the  section 
related  to  the  line  of  centers? 

Sug.  1.  Let  S  and  S^  represent  two  intersecting  spheres 
whose  centers  are  C  and  C.  Pass  a  plane  through  C  and  C. 
What  are  the  circles  formed  ? 

Sug.  2.  Letter  points  of  intersection  of  the  circles  P  and 
P'  and  draw  the  chord  P  P'. 

Extend  the  line  C  Q!  to  meet  the  circumferences. 

8ug.  3.     How  is  C  Q!  related  to  P  P?     Why? 

Sug.  4.  Imagine  the  lower  half  of  the  figure  composed 
of  the  intersecting  circles  to  revolve  about  the  axis  C  C  pro- 
duced. What  will  the  two  semicircles  generate?  the  line 
O  P?  the  point  P? 

Give  a  complete  demonstration,  and  write  Prop.  X. 

Exercises. 

512.  Three  equal  lines  have  their  extremities  in  the  sur- 
face of  a  sphere.  How  are  these  lines  related  to  the  center  of 
the  sphere  ? 

513.  If  a  cone  of  revolution  roll  upon  a  plane  with  its 
vertex  fixed,  what  kind  of  a  surface  is  generated  by  the  sur- 
face of  the  cone? 

Can  you  prove  that  one,  and  only  one,  surface  of  a  sphere 
may  be  passed  through  any  four  points  not  in  the  same  plane? 

D 


h 1> 

Sug.  1.     Suppose  A,  B,  C,  D  to  be  the  four  points  not  in 
he  same  plane. 


THE   SPHERE.  353 

What  is  the  locus  of  all  points  equally  distant  from  A 
and  B,  B  and  C?      What  is  the  intersection  of  these  two  loci? 

Stig.  2.  What  is  the  locus  of  all  points  equally  distant 
from  B,  D  ? 

Note  the  intersection  of  the  loci. 

The  pupil  will  work  out  the  demonstration. 


SPHERICAL  ANGLES  AND  POLYGONS. 
Definitions. 

638. 

T/ie  a7igle  of  two  i7itersecting  curves  is  the  angle  of  the 
two  tangents  to  the  curves  at  their  point  of  intersection.  This 
definition  applies  to  all  curved  surfaces. 

639. 

A  sf>herical  angle  is  the  angle  included  between  two  arcs 
of  great  circles  of  a  sphere.  The  arcs  are  the  sides  of  the  angle 
and  their  intersection  the  vertex. 

640. 

A  portion  of  the  surface  of  a  sphere  bounded  by  three  or 
more  arcs  oi  great  circles  is  called  a  spherical  polygon. 


The  sides  of  the  spherical  polygon  are  the  bounding  arcs; 
23 — 


354  SOLID  GEOMETRY,  BOOK  VlII. 

the  angles  of  the  polygon  are  the  angles  of  the  intersecting 
arcs;  the  vertices  of  the  polygon  are  the  points  of  intersection 
of  the  arcs. 

Thus,  A  B  D  K  is  a  spherical  polygon.  Its  sides  may  be 
expressed  in  degrees. 

641. 

The  diagonal o{  any  spherical  polygon  is  the  arc  of  a  great 
circle  joining  any  two  vertices  not  adjacent. 

642. 

The  planes  of  the  sides  of  a  spherical  polygon  form  2,  poly - 
edral  angle  whose  vertex  is  at  the  center  of  the  sphere. 

Thus,  C  —  ABDEisa  polyedral  angle  with  its  vertex 
ate. 

643. 

A  convex  spherical  polygon  is  a  spherical  polygon  whose 
corresponding  polyedral  angle  is  convex. 

644. 

A  spherical  triangle  is  a  spherical  polygon  of  three  sides. 

645. 

A  spherical  triangle  may  be  right,  acute,  equilateral, 
isosceles,  etc.,  under  the  same  restrictions  as  plane  triangles. 

646. 

Any  two  points  on  the  surface  of  a  sphere  may  be  joined 
by  two  arcs  of  a  great  circle  ;  one  will  usually  be  greater  than 
a  semicircumference,  the  other  less. 

The  smaller  arc  is  always  meant  unless  otherwise  stated. 


SPHERICAL  TRIANGLES  AND  POLYGONS. 


255 


647. 

Two  spherical  polygons  are  equal  when  one  may  be  ap- 
plied to  the  other  so  that  they  will  coincide  in  all  their  parts. 
Thus, 


the  spherical  triangles  ABC  and  A'  B'  Q!  are  equal  if  A  B, 
B  C,  C  A  are  equal  respectively  to  A'  B',  B'  C,  C  A',  and  the 
angles  A,  B,  C  respectively  equal  the  angles  A',  B',  C 


648. 

Two  spherical  polygons  are  symmetrical  when  the  sides 
and  angles  of  one  are  equal  respectively  to  the  sides  and  an- 
gles of  the  other  when  taken  in  reverse  order.  Study  the 
figures. 


Can  you  make  these  figures  coincide? 
[The  question  of  equality  and  symmetry  of  spherical  tri- 
angles may  be  cleared  by  using  the  rind  of  an  orange. 


356 


SOLID  GEOMETRY,  BOOK  VIII. 


lyet  the  pupil  draw  a  vSphere  and  show  that  the  vertices  of 
one  symmetrical  triangle  are  at  the  ends  of  the  diameters  from 
the  vertices  of  the  other.] 

649. 

What  is  meant  by  vertical  spherical  polyedral  angles? 
Draw  a  figure  to  illustrate.  How  are  the  corresponding  poly- 
gons of  two  vertical  polyedral  angles  related  to  each  other? 

650. 

A  polar  triangle  is  formed  by  taking  the  vertices  of  a 
given  spherical  triangle  as  poles,  and  then  describing  three 
intersecting  arcs  of  great  circles.     Thus,  in  the  figure. 


A  is  the  pole  of  B'  C; 
B  is  the  pole  of  A'  C; 
C  is  the  pole  of  A'  B'. 


651, 


The  great  circles  B'  B'^  C  C ,  B'  Q!  form  eight  spherical 
triangles,four  of  which  are  on  the  opposite  side  of  the  sphere. 
That  one  of  the  eight  is  a  polar  A  which  has  A'  homologous 
to  A  on  the  same  side  of  B  C,  C  and  C  on  the  same  side  of 
A  B,  B'  and  B  on  the  same  side  of  A  C. 


SPHERICAL  TRIANGLES. 


357 


652. 

Proposition  XI. 

(1)  In  the  two  given  As  let  A  B  C  and  A'  B'  C  be  sj^m- 
metrical isosceles  spherical  As;  i.  ^.,  A  B  =  A'  B',  A  C  =  A'  C', 
B  C  =  B'  C,  and  Z  A  =  Z  A',  Z  B  =  Z  B',  and  Z  C  =  Z  C. 

Can  they  be  made  to  coincide? 


[///«/.— Compare  A  B  and  A'  C,  A  C  and  A'  B'.     [Auth. 
Compare  Z  A  with  Z  A'.] 

Can  you  superpose  A  A  B  C  upon  A  A'  B'  C? 

Will  the  two  As  coincide.'*    Why? 

Give  a  complete  demonstration. 

(2)     When  the  two  triangles  are  not  isosceles. 

Can  you  show  how  they  are  related  ? 


I 

In   the  figure  let  A  B  C  be  any  spherical  triangle  and 
A'  B'  C  its  polar  triangle.     Draw  on  the  surface  of  a  sphere. 


858  SOLID  GEOMETRY,  BOOK  VIII. 

Can  you  prove  that  A  B  C  is  the  polar  triangle 

of  ^'  B'  C  ? 

Sug.  I.     A  is  the  pole  of  what  arc  ? 

How  many  degrees  from  A  to  B'  ? 

Sug.  2.     C  is  the  pole  of  what  arc  ? 

How  far  from  C  to  B'?    B'  is  the  pole  of  what  arc  ? 

Pupil  complete  proof. 

Write  the  theorem,  and  call  it  Prop.  XII. 

654. 
Proposition  XIII. 

In  the  figure  let  A  B  C  and  A'  B'  C  be  polar  triangles. 

Can  you  show  that  each  angle  of  one  is  meas- 
ured by  the  supplement  of  the  side  opposite  it  in 
the  other  ? 


Siig.  1 .     Select  any  angle  as  A',  and  extend  its  sides,  if 
necessary,  to  meet  the  opposite  side  of  the  other  triangle  at 
D  and  E.     What  part  of  a  circumference  is  the  distance  from 
each  point,  D  and  E,  to  the  extremities  of  the  opposite  sides  ? 
i.  <?.,  B  is  the  pole  of  what  arc  ?     E  is  the  pole  of  what  arc  ? 
Sng.  2.     D  C  +  B  E  =  ? 
BC+  DE  =  ? 
.-.  DE  =  ? 
Complete  the  demonstration,  and  write  Prop.  XIII. 


SPHERICAL   TRIANGLES. 

655. 

Proposition  XIV.  - 


In  the  given  sphere  form  any  spherical  triangle,  as  A  B  D, 
and  pass  planes  through  the  sides  and  the  center  of  the 
sphere.  What  is  the  figure  formed?  Compare  a  face  an- 
gle of  any  triedral  angle  with  the  sum  of  the  other  two. 
[§449.] 

Can  you  prove  that  the  sum  of  any  two  sides  of 
a  spherical  triangle  is  greater  than   the  third  side? 


656. 


Cor,  Can  you  prove  that  any  side  of  a  spherical 
triangle  is  greater  than  the  difference  of  the  other 
two  sides  ? 


860    .  SOLID  GEOMETRY.  BOOK  VIII. 

657. 

Proposition  XV. 

What  is  the  limit  of  the  sum  of  all  the  face  angles  about  a 
polyedral  angle  ?     [  §  534] 


In  the  figure  let  A  B  C  D  be  any  polygon  on  the  sur- 
face of  the  sphere.  Pass  planes  through  the  sides  and  the 
center,  C,  of  the  sphere.     What  is  formed  by  these  planes  ? 

Can  you  prove  that  the  sum  of  the  sides  of  any 
spherical  polygon  is  less  than  the  circumference  of 
a  great  circle? 

Give  demonstration,  and  write  Prop.  XV. 

658. 

Proposition  XVI. 


In  the   figure  let  A  B  C  be  any  spherical  triangle  and 
A'  B'  C  its  polar  triangle.     How  do  you  measure  each  angle  ? 


SPHERICAL   TRIANGLES. 


361 


Write  an  expression  for  the  measurement  of  Z  A  +   Z  B  + 
Z  C.     What  is  the  limit  of  c^  -^  b'  -\-  c' '>.     [Prop.  XV.] 

Can  you  now  demonstrate  that  the  sum  of  the 
angles  of  any  spherical  triangle  is  greater  than  two 
and  less  than  six  right  angles. 

659. 

Cor.  How  many  right  angles  may  a  spherical 
triangle  have?   how  many  obtuse  angles? 

660. 

A  biredayigular  spherical  triangle  is  one  which  has  two 
right  angles. 

A  trired angular  spherical  triangle  is  one  which  has  three 
right  angles. 

661. 

Proposition  XVII. 


Fig.  1.  Fig.  2. 

Let  ABC  and  A'  B'  C  be  two  symmetrical  spherical 
triangles  having  their  homologous  vertices  diametrically  op- 
posite to  each  other. 

Are  the  triangles  equivalent? 

Sug.  1.     Let   P  be   the   pole  of  a   small  circle  passing 


362  SOLID  GEOMETRY,  BOOK  VIII. 

through  the  points  A,  B,  C  and  let  P  O  P'  be  a  diameter. 
Draw  the  great  circle  aics  PA,  P  B,  P  C,  F  A',  P'  B',  P'  C. 

Sug.  2.     Compare  P  A,  P  B  and  P  C- 

Also  compare  P  A  and  P'  A';  P  B.  P'  B';  P  C,  P'  C 

Also  compare  P'  A',  P'  B',  and  P'  C;  then  compare  As 
P  A  C  and  P'  A'  C.  (1)  Are  they  symmetrical?  Are  they 
isosceles?     Draw  conclusion. 

Sug.  3.  In  the  same  manner  compare  (1)  As  P  A  B 
and  F  A'  B'.  (2)  As  P  B  C  and  V  B'  C. 

Sug.  4.  Now,  by  referring  to  the  three  pairs  of  As 
above,  compare  As  A  B  C  and  A'  B'  C. 

Sug.  5.  But  suppose  P,  the  pole  of  the  small  Q  passing 
through  the  points  A,  B,  C,  shall  fall  without  the  A  A  B  C. 

Can  you  show  that  each  of  the  two  As  A  B  C 
and  A'  B^C  is  equivalent  to  the  sum  of  two  isosceles 
As  diminished  by  the  third?   (See  second  figure  ) 

Draw  general  conclusion,  and  write  Prop.  XVII. 

662. 

Proposition  XVTII. 


Given:  On  the  same  or  equal  sphere  two  triangles  in 
which  two  sides  and  the  included  angle  of  one  equal  respect- 
ively two  sides  and  the  included  angle  of  the  other. 

Make  deductions  and  prove. 

Sug.  1.     In  Fig.  1  let  As  A  B  C  and  D  E  F  on  the  same 


SPHERICAL   TRIANGLES. 


Sphere  have  Z  A  and  the  sides  A  B,  A  C  of  the  one  respect- 
ively equal  to  Z  D  and  sides  D  E  and  D  F  of  the  other.  Can 
the  triangles  be  made  to  comcide  ? 

Sug.  2.  In  Fig.  2,  on  a  sphere  equal  to  that  in  Fig.  1 . 
redraw  A  A  B  C  and  draw  A  D'  E'  F'  symmetrical  to  A  D  E  F 
of  Fig.  1. 

Compare  As  D  E  F  and  D'  E'  F'.     [§  662.] 

Pupil  complete  demonstration,  and  write  Prop.  XVIII. 

663. 

Proposition  XIX. 


Given:  Two  triangles  on  the  same  sphere,  or  equal 
spheres,  having  two  angles  and  the  included  side  of  one  equal 
respectively  to  two  angles  and  the  included  side  of  the  other. 

(1)  Can  you  prove  the  triangles  equal? 

(2)  Can  you  prove  the  triangles  symmetrical  ? 
Study  the  figures  and  give  proof. 

664. 

Proposition  XX. 

Suppose  on  the  same  or  equal  spheres  the  three  sides  of 
one  triangle  equal  the  three  sides  of  another  respectively. 

Compare  the  triangles. 


364 


SOLID  GEOMETRY,  BOOK  VIII. 


In  the  figure  let  A  A   B  C  and  A  A'  B'  C  have  the  three 
sides  of  one  respectively  equal  to  the  three  sides  of  the  other. 


Stig.  1.  Connect  the  vertices  of  each  A  with  the  centers 
of  the  spheres. 

Su^".  2.  Compare  the  face  angles  of  the  triedral  Z  at 
O  with  the  face  angles  of  the  triedral  Z  at  O'.  Compare  the 
diedral  Z  s  'of  the  two  solid  angles. 

^Sug.  3.  Can  you  now  compare  A  A  B  C  and  A  A'  B'  C\ 
(1)  when  sides  are  arranged  in  the  same  order,  (2)  when  sides 
are  in  reverse  order  ? 

Write  Prop.  XX. 

665. 

Proposition  XXI. 


Given:     The  isosceles  spherical  Zi  A  B  C  with  A  B  =  A  C. 
Can  you  prove  z  B  =  z  C  ? 


POLAR  TRIANGLES.  365 

S2ig.  Pa<:s  the  arc  of  a  great  O  through  A  and  D,  the 
raid-point  of  B  C. 

Prove  the  As  equal  or  equivalent  and  consequently  equi- 
angular.    Give  proof  in  full. 

Write  Prop.  XXI. 

666. 

Proposicion  XXII. 


Given:  The  As  A  B  C,  A'  B'  C  on  the  same  or  equal 
spheres,  with  the  angles  of  the  one  respectiv  ly  equal  to  the 
angles  of  the  other. 

How  do  the  As  compare? 

Sug^.  1.  Construct  the  polar  As  T  and  T'  for  the  given 
As.     Compare  the  sides  of  T  and  T'.     (What  Prop.?) 

Compare  the  angles  of  T  and  T'.     (What  Prop.?) 

Sug.  2.  Compare  the  sides  of  the  As  A  B  C  and  A'  B'  C. 
Therefore  by  Prop.     —     —     —     — 

Write  Prop.  XXII. 

667. 

Cor.  /.  If  two  angles  of  a  spherical  triangle  are 
eqtial,  can  you  show  that  the  sides  opposite  these 
angles  are  eqtial,  and  the  triangle  isosceles  ? 

In  figure,  §666,  what  measures  Z  C?  Z  B?  Then  what 
sides  are  equal?  what  angles?  etc. 


866 


SOLID  GEOMETRY,  BOOK  VIII. 

668. 


r 

I 

^D 

b. 

\'!      / 

Cor.  IF.  In  the  figure  suppose  three  planes 
each  perpendicular  to  the  other  two  and  passing 
through  the  center  of  the  sphere.  Can  you  show  how 
many  equal  trirectangular  triangles  are  formed? 

669. 

Proposition  XXIII. 

I.  Given:  ABC  any  spherical  triangle  in  which  two 
angles  are  unequal;  Z  C  >   Z  A. 


How  do  the  sides  opposite  these  angles  compare? 
Which  is  the  greater  ? 

Sug.  1.     Draw  C  D  arc  of  a  great  O  so  that  Z  A  C  D 
shall  equal  Z  C  A  D. 

Compare  A  D  and  C  D. 

Compare  D  B  +  C  D  with  B  C.     [§  655.] 


EXERCISES.  367 

Compare  D  B  +  A  D  with  B  C. 


Pupil  give  complete  demonstration. 

11.  Suppose  the  sides  A  B  and  B  C  are  unequal,  A  B  > 
BC. 

How  do  the  angles  opposite  these  sides  com- 
pare ?     Which  is  the  greater  ? 

Use  method  of  exclusion.     Write  Prop.  XXIII. 
Exercises. 

514.  The  arc  of  a  great  circle  drawn  from  the  vertex  of 
an  isosceles  spherical  A  to  the  middle  point  of  the  base  is  1  to 
the  base  and  bisects  the  vertical  angle. 

515.  Suppose  one  circle  of  a  sphere  passes  through  the 
poles  of  another  circle  of  a  sphere,  how  are  the  two  circles 
related? 

516.  Compare  the  volume  formed  by  revolving  a  rect- 
angle about  its  shorter  side  with  that  formed  by  revolving 
it  about  its  longer  side. 

517.  If  the  sides  of  a  spherical  A  are  respectively  63°, 
115°,  and84°,  how  many  degrees  in  each  angle  of  the  polar  A  ? 

518.  If  the  angles  of  a  spherical  A  are  100°,  90°,  and  75°. 
how  many  degrees  are  there  in  each  side  of  its  polar  A  ? 

519.  What  are  the  maximum  and  minimum  limits  of  the 
sum  of  the  angles  of  a  spherical  pentagon  ? 

520.  At  a  given  point  in  a  given  arc  of  a  great  circle, 
to  construct  a  spherical  angle  equal  to  given  spherical  angle. 

521.  The  radius  of  a  small  circle  on  a  sphere  is  less  than 
the  radius  of  the  sphere. 


368 


SOLID  GEOMETRY,  BOOK  VIII. 


SPHERICAL  MEASUREMENTS. 

Definitions. 

670. 

A  lune  is  a  portion  of  the  surface  of  a  sphere  included  be- 
tween  two  semicircumferences  of  great  circles;   as  A  B  C  D. 


The  angle  of  the  lune  is  the 'angle  formed  by  its  bound- 
ing arcs;  as  Z  C  A  D. 

671. 

Lunes  on  the  same  or  on  equal  spheres,  having  equal  an- 
gles, may  be  made  to  coincide. 

672. 

A  spherical  wedge  or  ungula  is  the  solid  bounded  by  the 
lune  and  the  planes  of  its  sides;  as  A  O  B  C  D. 

The  lune  A  B  C  D  is  called  the  base  of  the  wedge  and  the 
diameter  A  B  is  the  edge.  * 

673. 

A  zone  is  a  portion  of  the  surface  of  a  sphere  included  be- 
tween two  parallel  planes.  The  circumferences  of  the  sec- 
tion.«  are  called  the  bases  of  the  zone.  The  distance  between 
the  bases  is  the  altitude  of  the  zone. 


SPHERICAL  MEASUREMENTS.  369 

674. 

A  zone  of  one  base  is  a  zone  one  of  whose  bases  is  tangent 
to  the  sphere. 

P 


? 

If  the  circle  PA  B  P'be  revolved  about  the  diameter  P  P', 
the  arc  A  B  will  describe  a  zone  and  points  A  and  B  will  de- 
scribe the  circumferences  of  the  bases  of  the  zone. 

675. 

The  surface  of  a  sphere  has  eight  trirectangular  triangles, 
and  it  is  convenient  to  divide  each  into  90  equal  parts,  called 
spherical  degrees.  The  surface  of  every  sphere  has  how  many 
spherical  degrees  ?  What  kind  of  a  triangle  is  a  spherical 
degree  ? 

676. 

Proposition  XXIV. 


In  the  figure  let  A  C  K  and  B  C  D  be  any  two  arcs  of 
24 — 


370  SOLID  GEOMETRY,  BOOK  VIII. 

great  circles  intersecting  at  point  C  on  the  hemisphere 
A  D  B  B  forming  tTie  two  vertical  spherical  triangles  A  C  B 
and  D  C  E. 

Can  you  show  that  the  sum  of  the  two  As  men- 
tioned is  equivalent  to  a  lune  whose  angle  equals 
Z  ACB? 

Sug.  1.  Complete  the  circles  of  arcs  ACE  and  BCD. 
What  is  CD  FE?     Why?     C  A  F  B? 

Sug.  2.  Compare  arcs  C  B  and  D  F,  A  C  and  E  F,  A  B 
and  E  D. 

Sug.  3.     Compare  As  D  E  F  and  A  C  B.     [§  664.] 

Now  compare  the  sum  of  A  C  B  and  D  C  E  with  C  D  E  F. 

Give  complete  demonstration,  and  write  Prop.  XXIV. 

677. 
Proposition  XXV. 


In  the  figure  let  P  and  P'  be  the  poles  of  a  sphere  and 
draw  the  arc  of  a  great  O  A.  F.     Let  P  A  P'  F  be  a  lune. 

(I)  Compare  the  area  of  the  lune  with  the  area  of  the 
sphere  when  the  angle  of  the  lune  and  four  right  angles  are 
commensurable. 

Sug.  1.  On  the  arc  A  F  measure  ofi"  equal  distances,  each 
of  which  is  the  common  unit  between  arc  A  F  and  the  cir- 
cumference A  H  G  F.  From  O,  the  center  of  the  sphere, 
draw  lines  to  the  points  of  division  on  the  arc  A  F. 


SPHERICAL  MEASUREMENTS.  371 

Sug.  2.     Suppose  the  unit  contained  in  A  O  F  «  times 

and  in  four  right  angles  b  times;  then  -. •  ==  ? 

4  rt.  Z  s 

Now  pass  planes  through  P  P'  and  the  points  of  division, 

A,  B,   C,  D,  E,  F.     How  is   the  surface  of  the  lune  divided  ? 

Compare  the  number  of  lunes  with  the  number  of  Z  s  at  O. 

^,  ,        surface  of  the  lune 

Then  the  -^ -^-r r =  ? 

surface  of  the  sphere 

Complete  the  demonstration. 

(2)  Compare  the  area  of  the  lune  with  the 
area  of  the  sphere  when  Z  A  O  F  and  four  right 
angles  are  not  commensurable. 

Sug.     Use  method  of  limits. 

678. 

Cor.  I,  Can  you  show  that  a  lune  contains  twice 
as  man}^  spherical  degrees  as  its  angle  contains  angu- 
lar degrees  .'* 

Sug.  How  many  spherical  degrees  in  the  su  face  of  a 
sphere  ? 

What  is  the  sum  of  all  the  angular  degrees  of  all  the 
lunes  converging  at  P  in  the  figure  ?  I^et  L  =  lune,  S  = 
surface  of  sphere,  A  =  angle  of  lune. 

Can  you  prove  that 

(1)  L  :  S  ::  A  :  360°? 

(2)  The  number  of  spherical  degrees  of  L  :  720°  : :  A  : 
360°. 

(3)  L  =  2A? 

679. 

Cor,  II.  Can  you  prove  that  lunes,  on  the  same 
or  on  equal  spheres  are  to  each  other  at  their 
angles  ? 


372  SOLID  GEOMETRY,  BOOK  VIII. 


680. 


The  spherical  excess  of  a  triangle  is  the  excess  of  the  sum 
of  the  angles  of  a  spherical  triangle  over  two  right  angles. 

The  spherical  excess  of  a  polygon  is  the  excess  of  the  sum 
of  the  angles  of  the  spherical  polygon  over  two  right  angles 
taken  as  many  times  as  the  polygon  has  sides  less  two. 

Thus,  if  a  polygon  has  n  sides,  it  has  7i  —  2  spherical  tri- 
angles. Its  spherical  excess  equals  the  sum  of  the  spherical 
excesses  of  its  triangles. 

681. 

Proposition  XXVI. 


In  the  figure  let  A  D  E  be  any  spherical  triangle.  Com- 
plete the  great  circle  of  which  D  E  is  an  arc  and  produce  the 
sides  E  A  and  D  A  till  they  meet  the  great  circle  D  E  B  C. 

Compare  the  number  of  spherical  degrees  in  tri- 
angle DAE  with   the  number  of  angular  degrees. 
Sug.  1.     What   are  D  E  B,  D  A  B,  and  D  C  B?     [Auth.] 

(1)  ADAE+ABAC  equivalent?     Its  angle? 

(2)  A  D  A  E  +  A  A  B  E  equivalent?     Its  angle? 

(3)  A  D  A  E  +  A  D  A  C  equivalent  ?     Its  angle  ? 
Sug.  2.     How  many  spherical  degrees  in   a  lune  whose 

angle  is  A?  in    a  lune  whose  angle   is  B?  in  a  lune  whose 
angle  is  C  ? 

Sug.  3.  In  (1),  (2),  (3),  how  many  times  extra  have  we 
used  the  A  D  A  E  in  taking  the  surface  of  the  hemisphere  ? 


SPHERICAL  MEASUREMENTS.  873 

How  many  spherical  degrees  in  a  hemisphere  ? 

Sug.  4.     2  ^'^  D  A  E  +  360°  equivalent  ? 

A  D  A  E  +  180°  equivalent  ZA+ZB+ZC. 

Why? 

.-.  A  D  A  Eis  equivalent  to  (Z  A+   Z  B  +  Z 

C  —  180°)  spherical  degrees. 

If  we  call  the  spherical  excess,  E,  then  A  D  A.  E  contains 

E  spherical  degrees. 

Go  over  these  hints  carefully  again  and  give  a  complete 

demonstration. 

Write  Prop.  XXVI. 

Scholium.     If  a  spherical  triangle  contains  120  spherical 

degrees,  or  c  spherical  degrees.it  means  that  the  surface  of  the 

•        ,.120  c        ^   ^  ^  r     r.       '     ^ 

triangle  is  ottTi  '  ^^  ^iTTiTi  ^'  ^"^  surface  oi  a  hemisphere. 

O  O  U  o  t)  U 

How  would  you  express  the  relation  of  a  spherical  trian- 
gle which  has  r  spherical  degrees  to  the  surface  of  a  sphere? 

Exercises. 

522.  How  many  spherical  degrees  in  a  spherical  triangle 
whose  angles  are  180°,  140°,  120°? 

523.  Compare  the  surface  of  a  spherical  triangle  whose 
angles  120°,  150°,  1»0°  with  that  of  a  sphere. 

524.  Each  angle  of  a  spherical  triangle  is  90°.  Com- 
pare the  triangle  with  the  surface  of  a  sphere. 

525.  The  angle  of  a  lune  i-;  60°.  What  part  of  the  sur- 
face of  a  sphere  is  it  ? 

526.  The  sides  of  a  spherical  A  are  respectively  75°, 
120°,  and  90°.  How  many  degrees  in  each  angle  of  its  polar 
triangle  ? 

527.  If  the  angles  of  a  spherical  A  are  respectively  100°, 
90°,  75'',  how  many  degrees  are  there  in  each  side  of  its  polar 
triangle  ? 

528.  The  sum  of  the  angles  of  a  spherical  pentagon  is 
greater  than  six.  and  less  than  ten,  right  angles. 

529.  The  sides  opposite  the  equal  angles  of  a  birectangu- 
lar  triangle  are  quadrants. 


874  SOLID  GEOMETRY,  BOOK  VIII. 

530  If  the  radii  of  the  bases  of  the  frustum  of  a  cone 
are  r  and  /  and  the  slant  height  is  a,  and  the  altitude  is  h, 
can  3'ou  show  that  S  =  tt  «  (r  -[-  r')  ? 

531.  The  volume  of  the  frustum  of  a  right  cone  is  V. 
Find  the  difference  between  the  volumes  of  the  circumscribed 
and  the  inscribed  regular  hexagonal  frustums. 

532.  A  cannon  ball  4  inches  in  diameter  weighs  8  lbs. 
What  is  the  cost  of  one  of  the  same  material  whose  diameter 
is  6  inches,  if  the  metal  is  worth  three  cents  per  pound  ? 

533.  The  height  of  the  frustum  of  a  cone  is  |  of  the  height 
of  the  entire  cone.  Compare  the  volume  of  the  frustum  and 
the  entire  cone. 

534.  What  are  the  dimensions  of  a  right  cylinder  ]  I  as 
large  as  a  similar  cylinder  whose  height  is  20  feet  and  whose 
diameter  is  10  feet? 

535.  The  total  surface  of  a  sphere  is  16  square  feet. 
What  is  the  number  of  square  feet  in  a  lune  of  the  same  sphere 
whose  angle  is  30°  ? 

536.  If  the  surface  of  a  lune  is  2  square  feet  and  the 
surface  of  the  sphere  is  18  square  feet,  what  is  the  angle  of 
the  lune? 

682 

A  spherical  segment  is  the  portion  of  a  sphere  included 
between  two  parallel  planes. 

683. 

A  spherical  sector  is  the  portion  of  a  sphere  generated  by 
the  revolution  of  a  circular  sector  about  a  diameter. 

I.  Thus,  let  P  P'  be  the  diameter  of  any  circle,  and  P  O  C 
a  circular  sector.  When  the  semicircle  P  B  P'  is  revolved, 
the  circular  sector  P  O  C  will  generate  a  spheiical  sector, 
whose  base  is  described  by  the  arc  P  C,  and  whose  conical 
surface  is  described  by  the  radius  O  C. 


SPHERICAL  MEASUREMENTS. 


375 


The  arc  P  C  describes  a  zone  which  is  called  the  base  of 
the  spherical  sector. 

II.  What  will  be  the  circular  sector  COB  describe  ? 
How  many  surfaces  bound  it  ?  What  do  you  call  the  surfaces 
described  by  O  C  and  OB?  C  B?  What  is  the  base  of  the 
spherical  sector  described  by  C  OB? 

684. 

Proposition  XXVII. 


A 

e/._.' 


Let  in  71  be  an  axis  and  A  B  any  line  in  the  same  plane 
making  any  angle  with  the  axis,  but  not  meeting  it. 

Project  the  extremities  and  mid-point  of  A  B  upon  the 
axis,  thus  fixing  points  C,  D,  and  H. 

If  the  line  A  B  be  revolved  about  the  axis  m  n,  what  sur- 
face will  be  generated? 

How  do  you  compute  the  surface? 

From  A  draw  a  1  to  B  D  meeting  it  at  G,  and  at  E  erect 


876 


SOLID  GEOMETRY,  BOOK  VIII. 


a  1  to  A  B  meeting  the  axis  m  n  in  'F.     Compare  As  A  B  G 
and  H  K  F. 


^^||,andAB.EH^EF.AG 


CD    EF. 


What  expresses  the  convex  surface  described  by  A  B  ? 
.  •  .  area  of  the  surface  described  by  A  B  =  C  D  *  2  tt  E  F. 
Complete  the  demonstration  and  state  Prop.  XXVI. 

685. 

Cor.  I.     If  A  B  is  parallel  to  the  axis  m  n^  what  is  gen- 
erated?    How  is  the  lateral  surface  computed  ? 

686. 

Cor.  11.     Suppose  the  point  A  to  lie  in  the  axis  m  n^ 

What  is  generated  by  the  revolution  of  A  B  about 
the  axis  m  nf 

Compute  the  lateral  surface. 

Sug,     Use  the  method  as  when  A  B  did  not  meet  m   n. 
AC==0. 

687. 

Proposition  XXVIII. 
K 


Given:  K  B  D  E  a  semicircle,  and  A  D  an  arc.  Revolve 
this  arc  about  the  axis  K  E.     What  is  generated  ? 

Suppose  the  arc  to  be  divided  into  any  number  of  equal 
parts,  as  A  B,  B  C,  C  D.    Draw  the  chords  of  these  arcs.    Pro- 


SPHERICAL  MEASUREMENTS.  877 

ject  the  extremities  of  these  chords  on  the  axis  K  E.  Call 
K  O,  the  radius,  R. 

Compare  the  chords  A  B,  B  C,  C  D. 

At  the  mid-points  of  these  chords  erect  Is  terminating  in 
the  axis,  K  E. 

Where  will  these  Is  meet? 

Compare  the  length  of  the  Is. 

Can  you  show  that  the  area  of  a  zone  equals  its 
altitude  by  the  circumference  of  a  great  circle? 

Siig.  1.  What  is  the  area  generated  by  the  chord  A  B? 
by  the  chord  B  C?  CD? 

SuiT,  2.  Can  you  express  the  area  generated  by  the  sum 
of  the  chords  in  one  equition? 

Sug,  3.  Suppose  the  arcs  are  bisected  and  chords  drawn 
as  before,  what  will  represent  the  area  generated? 

Sug.  4,  Eet  the  arcs  be  bisected  indefinitely;  what  does 
the  broken  line  A  B  C  D  =  ? 

Sug,  5.  What  does  the  sum  of  surfaces  described  by  the 
chords  ==  ? 

What  does  2  tt  L  O  =  ? 

Review  and  give  a  complete  demonstration.  Write 
Prop.  XXVIII. 

Scholium,  If  we  let  S  represent  the  surface  of  the  zone 
and  h  the  altitude  and  r  the  radius  of  sphere  on  which  the 
zone  lies.  Prop.  XXVIII  may  be  expressed  in  the  formula 
S=2  7rrh. 

688. 

Proposition  XXIX. 

Recall  the  definition  of  a  zone.  Can  you  think  of  a  zone 
whose  altitude  equals  the  diameter?  Make  a  drawing  to 
illustrate. 


878  SOLID  GEOMETRY,  BOOK  VIII. 

How  do  we  find  the  area  of  the  surface  of  a  zone. 

Can  you  show  how  to  'find  the  area  of  the  sur- 
face of  a  sphere? 

Can  you  prove  this  by  the  method  used  in  Prop.  XXIX? 

689. 

Cor.  1.  Can  you  express  the  area  of  the  surface 
of  a  sphere  in  terms  of  the  radius? 

Sug.  Instead  of  using  tt  D  as  the  circumference,  use  the 
equivalent  of  D. 

690. 

Cor.  II.  Can  you  show  from  Cor.  I.  that  the 
area  of  the  surface  of  a  sphere  equals  the  area  of 
four  great  circles  ? 

691. 

Cor,  III.  Can  you  show  that  the  area  of  the 
surface  of  a  sphere  equals  the  area  of  a  circle  whose 
radius  is  the  diameter  of  the  sphere? 

692. 

Cor,  IV.  Show  that  the  areas  of  the  surfaces  of 
two  spheres  are  to  each  other  as  the  squares  of  their 
radii,  or  as  the  squares  of  their  diameters. 

693. 

Cor,  V.  Can  you  show  that  the  area  of  a  spher- 
ical degree  equals     ^^o    ^ 


SPHERICAL  MEASUREMENTS. 


879 


694. 

Proposition  XXX. 


Let  the  fig^ure  represent  a  cube  circumscribed  about  a 
sphere  whose  radius  is  r.  Join  the  vertices  of  the  cube  with 
the  center  of  the  sphere.  Pass  a  plane  through  each  edge  and 
the  two  lines  which  join  its  ends  to  the  center.  What  solid  fig- 
ures are  formed  with  vertices  at  the  center  of  the  sphere? 
What  are  their  bases?  their  altitudes?  What  is  their  sum  ? 
How  does  it  compare  with  the  volume  of  the  sphere  ?  How 
do  you  find  the  volume  of  each  pyramid?  Now  at  points 
where  the  edges  of  the  pyramids  pierce  the  surface  of  the 
sphere  draw  tangent  planes. 

What  will  these  planes  do  to  the  cube  ?  Then  how  will 
the  new  circumscribing  solid  compare  with  the  cube  ? 

What  will  the  edges  made  by  these  tangent  planes  form  ? 

Pass  planes  through  these  edges  and  the  center  of  the 
sphere  as  before.  What  will  now  be  formed  ?  How  find  the 
volume  of  the  sum  of  all  these  pyramids? 

Now  call  the  volume  of  circumscribing  solid  V'  and  its 
surface  S',  and  the  volume  of  the  sphere  V  and  its  surface  S. 

Now  let  the  number  of  pyramids  be  indefinitely  increased 
by  passing  tangent  planes  to  the  sphere  at  the  points  where 
the  edges  of  the  pyramids  pierce  the  vsurface  of  the  sphere. 

What  does  S'  =  ?     What  does  S'  X  J  r  =  ? 


880  SOLID  GEOMETRY,  BOOK  VIII. 

What  does  V  =  ?  Is  V  =  S'  X  i  r  true  for  any  num- 
ber of  faces  ? 

Can  you  draw  the  conclusion  ? 

Write  a  statement  expressing  the  volume  of  a  sphere, 
and  call  it  Prop.  XXX. 

No/e.—Lftt  the  pupil  inscribe  a  sphere  in  a  pyramid  and 
give  a  demonstration  in  full  for  finding  the  volume  of  a  sphere. 
Try  it  again  by  starting  with  the  sphere  inscribed  in  an 
icosaedron. 

695. 

Show  that  V  =  i^  r^  or  l-^  Dl 
Su^.     Express  S  in  terms  of  r. 

696. 

Cor.  II.     Denote  the  volumes  of  two  spheres  by  v  and  v\ 
and  their  radii  by  r  and  /,  and  their  diameters  by  d  and  d! . 
Show  that  v\v'  =  ^  '.r'''  =  d''  :  d'\ 

697. 
Cor.  Ill,     Can  you  prove  that  the  volume  of  a 
spherical  pyramid  equals  the  product  of  its  base  by 
one-third  of  the  radius  of  the  sphere  ? 

698. 

Cor.  IV,  Can  you  prove  that  the  volume  of 
a  spherical  sector  equals  the  product  of  the  zone 
which  forms  its  base  by  one-third  of  the  radius  of 
the  sphere.^ 

Exercise. 

537.  If  r  =  radius  of  a  sphere,  C  =  circumference  of  a 
great  circle,  v  =  volume  of  a  spherical  sector,  k  and  S  =  the 
altitude  and  area,  respectively,  of  the  corresponding  zone, 
show  that  V  =  ^  IT  r"^  h. 


SPHERICAL  MEASUREMENTS.  381 

699. 
Proposition  XXXI. 

Draw  a  semicircle.  Take  any  arc,  A  B,  and  to  the  ex- 
tremities draw  radii  AC,  B  C,  and  draw  the  ±s  to  the  diam- 
eter, B  E  and  A  D. 


Let  this  figure  be  revolved  about  the  diameter  F  G. 

What  does  C  B  E  generate?     A  C  B?     A  CD? 

If  we  add  the  volume  generated  by  C  B  E  to  that  gen- 
erated by  A  C  B,  and  then  deduct  the  volume  generated  by 
A  C  D,  what  solid  will  remain  ? 

Can  you  show  how  to  find  the  volume  of  a  spher- 
ical segment? 

Suir.  1.  Call  the  radius  of  the  sphere  r,  radius  of  upper 
base  of  segment  /,  radius  of  lower  base  r'\  altitude  of  seg- 
ment D  E.  /i,  the  volume  of  the  segment  v. 

Sug,  2.  Find  an  expression  for  the  cone  generated  by 
CBE. 

What  represents  the  volume  of  the  sector  generated  by 
A  C  B? 

Find  an  expression  for  the  cone  generated  by  A  D  C. 


382  SOLID  GEOMETRY,  BOOK  VIII. 


Sug.  3.     Find  the  value  of  C  E  ia  terms   of  r   and   /, 
/^  =  E  C  —  D  C. 


Find  the  value  of  D  C    in  terms  of  r  and  r". 
Can  you  show  that 

(1)  z;  =  I-  TT  [2  r^  (C  E  —  C  D)   +  {r'  —  C^' )  C  E  —  (r^  — 

C^')CD]? 

(2)  z;=:    1  TT   [2  r^CE  -CD)+  r^  (C  E -- CD)  —  (ce' 

-CD^? 

(3)  z;  =  1  TT  /2  [3  r'  — (c  e'  +  C  E  •  C  D  +  "cl5')  ]  ? 

(4)  (c  E  —  C  D)'  =  C^'  =  2  C  E  •  C  D  -}-  C^'  =  h^  ? 


(5)  3CE   -I-3CD  r=/^2-^-2CE  +2CE-CD  +  2C  D? 

(6)  C^'  +  CE-  CD  +  C~D'=f  (C^'  +   Cli')— I? 

(7)  .-.  from  (3)  z;  =  J  TT  /^  [  f  (/^  +  /'^)  +  f  ]• 

Review  these  steps  carefully  until  you  thoroughly  under- 
stand this  proposition. 
Write  Prop.  XXXT. 


700. 

Cor,     If  the  segment  be  of  one  base,  as  that  gen 
erated  by  F  B  E,  show  that 


SPHERICAL  MEASUREMENTS.  383 

701. 
Proposition  XXXII. 


Given:  The  cylinder  A  E  B  C  D  with  the  inscribed 
sphere  H  K  F  G.  Call  A  K  or  O  K,  r,  the  radius  of  the  sphere 
and  the  radius  of  the  base  of  the  cylinder. 

Can  you  prove  (1)  that  the  surface  of  the  cyl- 
inder :  to  surface  of  the  sphere  :  :  3  :  2,  and  (2)  th^t 
the  volume  of  the  cylinder  :  the  volume  of  the 
sphere  :  :  3  :  2  ? 

Sug.  1.  Express  the  factors  in  terms  of  tt  and  A  D  or  in 
terms  of  tt  and  r. 

Note. — The  celebrated  geometer  Archimedes  discovered 
this  interesting  theorem.     Read  his  biography. 

702. 

Cor.  Suppose  a  cone  to  have  the  same  base  and  altitude 
as  the  cylinder  circumscribing  the  sphere. 

Can  3/0U  show  that  the  cylinder  :  sphere  :  cone 
::  3  :  2  :  1  .? 

Sug.     Express  volume  of  cone  in  terms  of  tt  and  r. 

Exercises. 

538.  What  is  the  convex  surface  of  the  largest  cylinder 
that  can  be  made  from  a  cube  whose  edge  is  14  feet  ? 


384  SOLID  GEOMETRY,  BOOK  VIII. 

539.  A  cube  of  steel  weighs  9  pounds.  What  will  be  the 
largest  bicycle  cone  that  can  be  turned  from  it?  1  cubic  inch 
of  steel  weighs  4.53  +  ounces. 

540.  If  the  edge  of  a  regular  tetraedron  is  4,  can  you 
show  that  the  radius  of  the  inscribed  and  circumscribed 
spheres  equals  ys  V^6~and  V^"?  Compare  the  volume  of  a 
cube  inscribed  in  a  sphere  with  that  circumscribed  about  the 
sphere. 

541.  Let  an  equilateral  triangle  revolve  about  an  alti- 
tude. Compare  the  convex  surface  of  the  cone  generated 
with  the  surface  of  the  sphere  generated  by  the  inscribed 
circle. 

542.  In  Ex.  541  compare  the  volumes  generated. 

543.  Given  a  cone  the  radius  of  whose  base  equals  the 
radius  of  a  sphere,  and  whose  altitude  equals  the  diameter  of 
the  sphere.  Can  you  prove  the  volume  to  each  other 
asl  :2? 

544.  On  the  same  sphere,  or  on  equal  spheres,  zones  of 
equal  altitudes  are  equal  in  area. 

545.  How  many  square  feet  in  a  spherical  triangle  whose 
angles  are  200°,  156°,  95°,  the  radius  of  the  sphere  being  15 
inches  ? 

546.  How  many  sheets  of  tin  20  inches  by  28  inches  are 
required  to  cover  a  globe  32  inches  in  diameter. 

547.  Compare  the  volume  of  the  moon  with  that  of  the 
earth,  assuming  the  diameter  of  the  moon  to  be  2,000  miles 
and  that  of  the  earth  8,000  miles. 

548.  What  is  the  cost  of  cementing  the  bottom  and 
curved  surface  of  a  cylindrical  cistern  10  feet  deep  and  8  feet 
in  diameter,  at  20  cents  per  square  yard  ? 

549.  What  is  the  ratio  of  the  surface  of  a  sphere  to  the 
entire  surface  of  its  hemisphere  ? 

550.  From  Ex.  547  compare  the  amount  of  light  reflected 
to  a  given  point  in  space  equally  distant  from  both  the  earth 
and  moon. 


EXERCISES.  385 

551.  Prove  how  the  areas  of  two  zones  on  the  same  or 
equal  spheres  are  related. 

552.  I,et  /  and  /'  be  the  radii  of  two  spheres.  How  are 
they  related? 

553.  The  altitude  of  two  zones  on  a  given  sphere  are  3 
inches  and  8  inches.     What  is  the  ratio  oT  their  surfaces? 

554.  What  is  the  polar  of  a  trirectangular  triangle? 

555.  A  spherical  triangle  is  to  the  surface  of  a  sphere  as 
the  spherical  excess  is  to  eight  right  angles. 

556.  All  triangles  on  the  sime  or  equal  spheres  having 
equal  angle-sums  are  equivalent. 

5o7.  What  is  the  volume  of  a  spheiical  segment  of  one 
base,  whose  altitude  is  6  cm.  and  the  radius  of  whose  sphere 
is  20  cm. 

558.  Compare  the  surface  of  a  sphere  of  diameter  d  with 
the  convex  surface  of  a  circumscribed  cylinder. 

559.  A  circular  sector  has  its  central  angle  30°  and  ra- 
dius 12  dm.  If  this  sector  is  revolved  about  a  diameter  per- 
pendicular to  one  of  its   radii,  find  the  volume  generated. 


25- 


386  FORMULAS  AND  NUMERICAL  TABLES. 


FORMULA.S  FROM  PREVIOUS  PROPOSITIONS. 

b  =  base. 

r  =  radius. 

c  =  circumference. 

r'  =  radius  upper  base. 

r"  =  radius  lower  base. 

s  =  slant  height. 

d  =  diameter 

h  =  altitude. 

a  =  apothem. 

S  =  surface. 

p  =  perimeter. 

A  =  area. 

a  ==  arc. 

V  =  volume. 

E  =  spherical  excess. 

T  —  trirectangular  triangle. 
Polygons: 

Rectangle,  h.  =  b  h.     §  251. 

Parallelogram,  K=  bh.     §  260. 

Triangle.  K  =  \  b  h.     §  261. 

Trapezoid,  K  =  \{b  ^  b')h.     §  265. 

Regular  polygons,  k.  =  \  a  p.     §  343. 

Circle,  C  =  27rr  =  7r^.     §  360. 

A  =  1  f  r  =  TT  r^  =  i  TT  a?'.     §  362. 
Polyedrons. 

Prism,  V  =b  h.     §  492. 

Lateral,  A  =/>>^.     §  473. 


FORMULAS  AND  NUMERICAL  TABLES.  387 

Parallelopiped,  v  =  b  h.     §  488. 
Pyramid,  v  =  \  b  h.     §  515. 

Lateral,  A  ==  J/)  .y.     §501.  

Frustum  of  pyramid,  v  ^  \  h  {b  ^  b'  -\-   ^b  b').  §519. 

Lateral  area,  K  =  \s.(p  ^  p').     Ex. 468. 
Cylinder,  v  ^ -k  r^  h.     §  571. 

Lateral  area,  K  ==  2  ir  r  h.     §  567. 

Entire  area,  A  =  2  tt  r  (r  +  h).     Ex.  497. 
Right  circular  cone,  v  =  \  -rr  r^  k.    §  600. 

Lateral  area,  A  =  tt  r  j.    §  596. 

Frustum,  v  =  \  ir  h  {r''  -\-  -/^  -{-  r  r')     §  602. 
Lateral  area,  A  =  tt  j  (r  +  r').    §  602. 
Sphere,  K  =  c  d  =  i  tt  r'  =  ir  d\   §§  687,  689. 
^  =  lrA==J7r^^  =  ^7r^^    §  694. 
Lune,  A  ==  2  a  T.    §  678. 
Spherical  triangle,  A  =  E  •  T. 
Spherical  polygon,  A  =  E  *  T. 
Zone,  A  =  2  TT  r  ^. 

Spherical  sector,  z;  =  |  tt  r*  ^.    §  697. 
Spherical  segment,  v  =  ^  w  A  (/  -\-  r")  -\-  \  tr  h^. 


Useful  Numej 

RicAL  Results. 

v/  2  =  1.41421. 

V  2  =  1.2599'e. 

VX=  1.73205. 

V  y  =  1.44224. 

V  5  ==  2.23606. 

Vt=  1.5874. 

v/  6  ==  2.4495. 

V  5  =  1.70998. 

v^  7  =  2.64575. 

^"7"=  1.91293. 

v^  10  =  3.1623. 

v^  9  =  2.0801. 

v' J  =  0.7071. 

ViO  =2.1544. 

"     1  decimeter, 

*'      dm. 

"     1  meter, 

"       m. 

"     1  dekameter, 

Dm. 

'*     1  hektometer. 

"       Hm. 

"     1  kilometer. 

Km. 

"     1  myriameter, 

'       Mm. 

388  TABLES  OP  DENOMINATE  NUMBERS. 

METRIC  MEASURES. 
Linear  Mkasurk. 

10  millimeters,  marked  mm.,  are  1  centimeter,  marked  cm. 

10  centimeters 

10  decimeters 

10  meters 

10  dekameters 

10  hektometers 

10  kilometers 

Rem. — The  measures  chiefly  used  are  the  meter  and  kil- 
ometer. The  meter,  like  the  yard,  is  used  ia  measuring  cloth 
and  short  distances;  the  kilometer  is  used  in  measuring  long 
distances. 

lm.=  39.37  in. 
1  km.  =  .6214  mi. 

Square  Measure. 

100  cq.  decimeters  =  |  I  'I'  ™^*^f-    , 
^  /  1  centare  (ca.). 

100  centares  =  1  are  (a.). 

100  ares  =  1  hektare  (Ha.). 

I  sq.  m.  =  1.196  sq.  rds. 

1  are        =  3.954  sq.  yds. 

1  Ha.       =  2.471  acres. 

Volume. 

1000  cu.  mm-  =  1  cu.  cm. 

1000  cu.  cm.    ==  1  cu.  dm. 

1000  cu.  dm.    =  1  cu.  m.  =  1  stere. 

1  cu.  cm.  =    .061  cu.  in. 

1  cu.  m.    =  1.308  cu.  yd. 

1  stere       =     .2759  cord. 

Capacity  Table. 

10  centiliters,  marked  cl.,  are  1  deciliter,  marked  dl. 

10  deciliters  "     1  liter,  "  1. 

10  liters  ''     1  dekaliter,      "  Dl. 

10  dekaliters  **     1  hektoliter.    "  HI. 

1  liter  =  1  cu.  dm.  =  1.0567  qt. 


TABLES  OF  DENOMINATE  NUMBEBS.  389 

MEASURES  OF  WEIGHT. 

The  gram  is    the  unit  of  weight;   it    is  legal  at  15.432 
grains. 

Table. 

10  milligrams,  marked   mg..    are  I  centigram,  marked  eg. 
10  centigrams  "     I  decigram,         "      dg. 

I/'VJ-^* a  1     (<  


10  decigrams 
U)  grams 
10  dekagrams 
10  hektograms 
10  kilograms 


1  gram,  ''  g. 

1  dekagram,  "  Dg. 

1  hektogram,  "  Hg. 

1  kilogram,  "  Kg. 

1  myriagram,  "  Mg. 

1  quintal,  *'  Q. 

metric  ton,  "  M.  T. 


10  myriagrams  "     1 

10  quintals,  or  1000  kilograms,  "      1 

1  gram  =  IT)  482  grains. 

1  Kg.  =    2.204B  lbs. 

1  tonneau     =    1.1023  tons. 

ENGLISH  MEASURES. 

DRY  MEASURE. 

Dry  measure  is  used  in  measuriig  grain,  vegetables,  fruit, 
coal,  etc. 

Table. 

2  pints  (pt.)  make  1  quart,  marked  qt. 
8  quarts  "        I   peck,         "        pk. 

4  pecks  "       1   bushel,      "        bu. 

Rem. — The  staridard  unit  of  dry  measure  isthe;  bushel 
it  is  a  cylindrical  measure  18|  inches  in  diameter,  8  inches 
deep,  and  contains  21501  cubic  inches. 

1  bu.  =    .3524  HI. 

1  dry  gallon  =  4.404  liters. 

LIQUID  MEASURE. 

Liquid  measure  is  used  for  measuring  all  liquids. 

Table. 

4  gills  (gi.)  make  1  pint,  marked  pt. 
2  pints  "       1  quart,       "         qt. 

4  quarts  "       1  gallon,     "         gal. 

Rem. — The  standard  unit  of  liquid  measure  is  tho^  gallon, 
which  contains  231  cubic  inches,  =  3.785  liters. 


SgO  TABLES  OP  DENOMINATE  NUMBERS. 

COMPARATIVK  TabIvE  OF  MKASURKS  OF  CAPACITY. 
I.IQUID  MEASURE.  DRY  MEASURE. 

1  gallon  =  281    cu.  in.  268|  cu.  in. 

1  quart    =    57|  cu.  in.  67|^  cu.  in. 

1  pint      =    28J  cu.  in.  38|  cu.  in. 

Measures  of  Weight. 

Troy  weight  is  used  in  weighing  gold,  silver  and  jewels. 

24  grains  (gr.)    make  1  pennyweight,  marked    pwt. 
20  pennyweights  "        I  ounce,  "  oz. 

12  ounces  "       1  pound,  "  lb. 

The  standard  unit  of  all  weight  in  the  United  States  is  the 
Troy  pound,  containing  5760  grains. 

Avoirdupois  Weight. 

16  ounces  (oz.)        r=z  1  pound.  marked  lb. 

100  pounds  =  1  hundredweight,        "        cwt. 

20  hundredweight  —  1  ton,  "       T. 

1  lb.  =   7000  grains  =  .4586  kilo. 

1  T.  =  .9071  tonneau. 

Long  Measure. 

Long  measure  is  used  it  measuring  distances,  or  length, 
in  any  direction. 

Table. 

12  inches  (in.)  make  1  foot,    marked   ft. 

3  feet                                          "     1  yard,  ''         yd. 

5J  yards,  or  16  \  feet,               "     1  rod,  *'         rd. 

820  rods                                         "     1  mile  *'         mi. 

Rem. — The  standard  unit  of  length  is  the  yard.  The  standard 

yard  for  the  United  States  is  preserved  at  Washington.  A  copy  of  this 
standard  is  kept  at  each  State  capital. 

1  yard  =    .9144  m. 
1  mile  =  1.6098  Km. 


TABLES  OP  DENOMINATE  NUMBERS.  391 

Square  Measure. 

144    square  inches  make  1  square  foot,  marked  sq.  ft. 

9    square  feet  "       1  square  yard,         "  sq.  yd. 

30J  square  yards        "       1  square  rod,  *'  sq.  rd. 

160    square  rods         "      1  acre,  "  A. 

640    acres  "      1  square  mile,         "  sq.  mi. 

1  square  yard  ==:  .8361  sq.  m. 
1  A.  =  .4047  hectare. 

Cubic  Measure. 

1728  cubic  inches  (cu  in.)  make  1  cubic  foot,  marked  cu.  ft. 

27  cubic  feet  "      1  cubic  yard,       "       cu.  yd. 

128  cubic  feet  (8  X  4  X  4),  8  ft.   longr,  )  .  ^^   ,  ,.        ^ 

4  ft.  wide,  and  4  ft.  high,  make  J  ^  ^°^^'  ^' 

Rem.— A  cord  foot  is  1  foot  in  length  of  the  pile  which  makes  a 
cord.  It  is  4  feet  wide,  4  feet  high,  and  1  foot  long;  hence  it  contains 
16  cubic  feet,  and  8  cord  feet  make  1  cord. 

1  CU.  yd.  ==  .7646  cu.  m. 
1  C   =  8.625  steres. 

Circular  Measure. 

60  seconds  (")  make  1  minute,  marked  '. 
60  minutes  "       1  degree,         "      °. 

360  degrees  "       I  circle. 

Rem. — The  circumference  is  also  divided  into  quadrants  of  90  de- 
grees each,  and  into  signs  of  30  degrees  each. 

Note. — 1.  A  degree  at  the  equator,  also  the  average  degree  of  lat- 
itude, is  equal  to  69.16  statute  miles. 

2.     Minutes  on  the  earth's  surface  are  called  geographic  miles. 


392  BIOGRAPHICAL  NOTES. 


A   BRIEF   ACCOUNT   OF    GEOMETRY  AMONG   THE 
BABYLONIANS,  EGYPTIANS,  AND  GRECIANS. 

Nearly  all  of  the  ancient  nations  leave  some  traces  of  a 
knowledge  of  the  simpler  notions  of  geometry.  It  is  said  that 
besides  the  kn'>wledge  of  the  division  of  the  circumference 
into  six  equal  parts  by  the  radius,  the  Babylonians  knew  some- 
thing of  the  properties  of  the  triangle  and  quadrangle.  Like 
the  Hebrews  (I.  Kings  vii.  23),  they  took  tt  =  3.  There  are 
evidences  that  the  Babylonians  knew  something  of  astronomy. 

Herodotus  and  other  ancient  historians  state  that  geom- 
etry had  its  origin  in  Egypt.  The  measurement  of  land  and 
the  building  of  the  pyramids  are  referred  to  by  some  of  the 
ancient  historians  as  evidences  of  their  knowing  some  practi- 
cal geometry. 

Before  1700  B.  C,  there  was  a  mathematical  manual, 
called  the  Papyrus,  containing  problems  in  arithmetic  and 
geometry.  It  was  written  by  Ahnies.  The  Egyptians  knew 
how  to  compute  the  area  of  an  isosceles  triangle  and  also  that 
of  an  isosceles  trapezoid.     Their  value  of  tt  was  3.1604. 

It  is  probable  that  the  Egyptians  knew  how  to  construct 
a  right  triangle  with  the  lines  whose  ratios  are  3:4:5.  Later 
geometricans  proved  that  some  of  the  rules  used  by  he  Egyp- 
tians in  their  constructions  were  not  absolutely  correct.  They 
did  not  have  a  system  of  geometry  based  on  a  few  axioms  and 
postulates. 

About  700  B.  C,  an  active  commercial  intercourse 
sprang  up  between  Greece  and  Egypt,  and  as  a  result  an  in- 
terchange of  ideas  arose.  Thales,  Pythagoras,  Plato,  and 
many  other  philosophers  visited  Egypt  to  study  the  learning 
of  that  nation. 


BIOGRAPHICAL  NOTES.  393 

Many  things  in  Greek  culture  originally  came  from  the 
land  of  the  pyramids.  The  Grecians  radically  changed  Egyp- 
tian geometry  and  the  subject  began  to  take  more  of  thr  form 
of  a  science. 

Thales  is  supposed  to  have  created  the  geometry  of  lines 
in  an  abstract  character.  He  formulated  many  of  the  truths 
which  the  Egyptians  were  acquainted  with.  Thus,  Eudemus 
ascribes  to  Thales  the  theorems  on  the  equality  of  vertical  an- 
gles, equality  of  the  angles  at  the  base  of  an  isosceles  triangle, 
the  bisection  of  a  circle  by  any  diameter,  and  the  equality  of 
any  two  triangles  having  a  side  and  any  two  adjacent  angles 
equal  respectively. 

Thales  was  one  of  the  earliest  geometers  to  apply  theo- 
retical geometry  to  practical  uses.  It  is  said  he  measured  the 
distances  of  ships  from  shore  by  geometry  As  a  result  of  his 
study  of  geometry  he  calculated  eclipses. 

Among  other  Grecians  who  added  to  the  science  of  geom- 
etry are  Pythagoras,  Hippocrates,  and  Anaxagoras.  For 
about  400  years,  650  B.  C.  to  250  B.  C,  the  science  of  geome- 
try had  many  of  its  principles  worked  out.  It  is  called  the 
golden  age  of  geometry.  The  great  body  of  the  propositions 
in  plane  geometry  is  about  as  it  was  formulated  by  Euclid. 
Many  improvements  in  methods  have  been  made  in  more  re- 
cent times,  such  as  the  study  of  loci,  coUinearity,  concurrence, 
and  other  subjects  in  what  is  called  Modern  Geometry.  These 
subjects  are  studied  in  colleges  and  universities. 
Brief  Biographies. 

Note. — Nearly  all  the  persons  mentioned  in  this  table 
have  their  names  connected  in  some  way  with  important 
principles  in  geometry.  The  numbers  refer  to  the  particular 
section  in  which  the  person  mentioned  has  made  some  dis- 
covery or  improvement,  b  stands  for  born,  d  for  died,  and  c 
(circa)  about. 

AhmeSy  c  1700  B.  C,   an  Egyptian  priest,  was  one  of 
the  earliest  writers  on  mathematics.     His  work  is  called  "Di- 
26 — 


394  BIOGRAPHICAL  NOTES. 

rections  for  Knowing  AH  Dark  Things."  It  is  a  collection  of 
problems  in  arithmetic  and  geometry.  He  gives  the  answers 
to  the  problems,  but  usually  does  not  show  the  processes  used 
to  obtain  them. 

In  one  arithmetical  problem  he  states  that  the  sum  of 

11,1  ,1.2 

—  —  ,  and  —    IS  — • 

24' 58  174'  23-45         29 

He  had  some  idea  of  algebra  and  always  used  for  the  un- 
known quantity  the  symbol  meaning  a  heap.  He  represented 
addition  by  a  pair  of  legs  walking  forward,  and  subtraction  by 
a  pair  of  legs  walking  backward,  or  by  a  flight  of  arrows. 
Equality  he  represented  by  the  sign  IL. 

In  the  part  treating  of  geometry  he  gives  the  contents  of 
barns;  but  since  he  does  not  give  the  shape  of  the  Egyptian 
barn,  we  cannot  verify  his  results.  He  finds  the  areas  of  rec- 
tilinear figures  and  of  a  circular  field  of  diameter  12.  The 
value  of  TT  he  approximates  closely  3.1416. 

[See  "A  Short  History  of  Mathematics,"  by  W.  W.  R. 
Ball,  pp.  3  to  8.] 

Anaxagoras,  c  450  B.  C,  tried  to  find  the  side  of  a 
square  which  should  equal  that  of  a  given  circle.  Anaxagoras 
belonged  to  the  Pythagorean  school  of  philosophy. 

Archimedes,  c  290—2 1 5  B.  C.  Born  at  Syracuse, educated 
at  Alexandria,  where  the  famous  Euclid  had  attended  fifty 
years  before.  Archimedes  was  one  of  the  earliest  writers  on 
measuring  the  circle.  He  proves  that  the  area  of  a  circle  equals 
that  of  a  right  triangle  having  the  circumference  for  the 
length  of  its  base,  and  the  radius  for  its  altitude.  He  assumes 
that  there  is  a  straight  line  equal  in  length  to  the  circumfer- 
ence. He  showed  that  this  line  exceeds  three  times  the  diam- 
eter by  a  part  which  is  less  than  \  but  more  than  \^  of  the  diam- 
eter.    To  quote  Ball's  "History  of  Mathematics"  : 

"In  the  old  and  mediaeval  world  Archimedes  was  unani- 
mously reckoned  as  the  first  of  mathematicians;  and  in  the 


BIOGRAPHICAL  NOTES.  395 

modern  world  there  is  no  one  but  Newton  who  can  be  com- 
pared with  him." 

His  mechanical  ingenuity  was  astonishing.  He  invented 
the  Archimedian  screw,  used  to  pump  water  out  of  the  hold 
of  a  ship  and  to  drain  the  fields  inundated  by  the  Nile.  He 
invented  engines  of  war  to  fight  the  Romans.  A  story  is  told 
of  his  burning-glass,  which  consisted  of  concave  mirrors,  a 
hexagon  surrounded  by  24-sided  polygons ;  this  he  used  to 
set  the  Roman  ships  on  fire. 

Euclid  wrote  systematic  treatises,  while  Archimedes 
wrote  brilliant  essays  addressed  to  famous  mathematicians  of 
his  day.  On  some  of  these  he  played  practical  jokes  by  mis- 
stating the  results,  "to  deceive  thOvSe  vain  geometricians  who 
say  they  have  found  everything,  but  never  give  their  proofs." 

On  Plane  Geometry  Archimedes  wrote  three  works,  viz. : 
(1)  •  The  Measure  oithe  Circle'';  (2)  'The  Quadrature  of  the 
Parabola";     (3)  ''Spirals:' 

On  Solid  Geometry  he  wrote:  (1)  "The  Sphere  and  the 
Cylinder";  (2)  "The  Conoids  and  Spheroids."  He  wrote 
a  treatise  on  the  thirteen  semi-regular  polyedrons^  solids 
bounded  by  regular  but  6Ss>^\m\\2iX  polygons. 

He  wrote  a  treatise  on  Arithmetic  in  which  he  calculates 
the  number  of  grains  of  sand  required  to  fill  the  universe  is 
less  than  10«^ 

He  wrote  a  treatise  on  Mechanics ;  another  on  Levers, 
in  which  he  declared  he  could  move  the  whole  earth  if  he  had 
a  fixed  fulcrum. 

He  wrote  a  work  on  floating  bodies,  and  while  bathing 
discovered  a  method  to  prove  that  the  crown  of  Hiero  was 
not  pure  gold. 

When  Syracuse  was  taken,  he  was  killed  by  a  Roman  sol- 
dier. It  is  said  that  he  was  contemplating  a  geometrical  fig- 
ure drawn  on  the  floor  in  the  sand  when  the  soldier  entered  and 
was  ordered  off  the  figure  by  Archimedes,  who  was  afraid  he 
would  spoil  it.    The  Roman  general,  Marcellus,  did  not  desire 


896  BIOGRAPHICAL  NOTES. 

the  death  of  Archimedes,  and  had  given  orders  to  spare  his 
house  and  person. 

The  Romans  built  a  splendid  monument  over  his  grave, 
upon  which,  according  to  his  wish,  was  engraved  a  sphere 
inscribed  in  a  cylinder  in  memory  of  his  discovery  that  the 
inscribed  sphere  is  two-thirds  of  the  cylinder,  and  that  the  area 
of  the  surface  of  the  sphere  equals  the  area  of  four  great  circles. 
[§§123,701.] 

Descartes  (1596 — 1650)  was  a  French  philosopher  as  well 
as  a  great  mathematician.  Descartes  was  among  the  first  to 
apply  algebra  to  geometry.  His  great  work  was  founding 
the  science  of  Analytic  Geometry. 

Buclid,  c  300  B.  C.  Euclid's  greatest  activity  was  in  the 
reign  of  the  first  Ptolemy,  306 — 283  B.  C.  Euclid  was  a  student 
of  the  Platonian  philosophy,  and  an  eminent  writer  on  geometry. 
He  gave  to  the  world  the  first  scientific  text-book,  ''Elements," 
in  thirteen  books,  which  is  still  a  standard  text-book  in  many 
English  schools.  Ptolemy  once  asked  Euclid  if  geometry 
could  not  be  mastered  by  an  easier  process  than  by  studying 
his  ** Elements."  The  answer  was,  ''There  is  no  royal  road  to 
geometry."  §  225  is  the  47th  Prop,  in  Euclid's  "Elements," 
and  is  often  referred  to  as  the  47th  of  Euclid. 

Kuler,  b  1707,  d  1783,  one  of  the  greatest  of  modern  math- 
ematicians. He  was  a  Swiss.  He  solved  in  three  days  math- 
ematical problems  which  eminent  mathematicians  had  said 
would  require  months.  His  intense  study  caused  him  to  go 
blind  and  he  dictated  his  '' Elements  of  Algebra'^  to  his  servant, 
who  was  quite  ignorant  of  mathematics.  This  work  is  still 
considered  one  of  the  best  of  its  class.  Besides  making  many 
discoveries  in  geometry,  he  did  much  work  in  higher  math- 
ematics and  astronomy.     [§§  277,  540.] 

Hero,  ^  1 90  B.  C.  He  was  a  practical  surveyor  of  Alex- 
andria and  made  many  mechanical  inventions,  among  them 
being  an  instrument  resembling  a  modern  theodolite.  This 
mathematician  wrote  a  commentary  on  Euclid's  "Elements." 


BIOGRAPHICAL  NOTES.  397 

Among  other  formulas,  he  developed  the  one  which  expresses 
the  area  of  a  triangle  in  terms  of  its  sides. 

Johannes  Kepler,  b  15"1,  d  1630,  was  a  great  student 
of  science  and  many  publications  are  from  his  pen.  He  has 
enriched  pure  mathematics  as  well  as  astronomy. 

He  was  one  of  the  first  great  mathematicians  to  make 
extensive  use  of  logarithms.  He  conceived  the  circle  to  be 
composed  of  an  infinite  number  of  triangles  with  their  com. 
men  vertices  at  the  center  and  their  bases  in  the  circumfer- 
ence, and  the  sphere  to  consist  of  an  infinite  number  of  pyra- 
mids. He  made  a  study  of  the  ellipse,  parabola,  and  hyper- 
bola; he  gave  to  the  world  his  laws  concerning  the  movements 
of  heavenly  bodies. 

I/Cgendre  (le  zhondr), /^  1752,  ^^1833,  one  of  the  most 
eminent  of  modern  mathematicians.  Besides  many  valuable 
additions  to  higher  mathematics,  he  gave  to  the  world  one  of 
the  most  celebrated  works  on  geometry.  It  is  called  the 
"Elements  de  Geometric."     [>5  ^05.] 

Sir  Isaac  Newton  (1642 — 1727)  was  a  great  English 
philosopher  and  mathematician.  Newton  was  a  student  of 
geometry,  but  his  works  are  on  higher  mathematics,  and  appli- 
cations of  algebra  and  geometry  to  the  solution  of  astronom- 
ical problems.     He  discovered  the  Biyiomial  Theorejn. 

Blaise  Pascal,  b  1623,  d  1662,  lived  in  Paris,  where  his 
father  taught  him.  The  father  would  not  trust  his  son's 
education  to  others.  Blaise  Pascal's  genius  in  geometry  showed 
itself  when  he  was  only  twelve  years  old.  The  father  tried  to 
keep  mathematical  work  from  his  son  till  he  had  learned  Latin 
and  Greek,  but  with  charcoal  and  paving  tiles  the  boy  stud- 
ied the  methods  of  drawing  the  circle  and  the  equilateral 
triang^le. 

He  discovered  for  himself  the  sum  of  the  three  angles  of 
a  triangle.  Pascal's  genius  was  so  great  for  geometry  that  at 
sixteen   he  wrote  a   treatise  on  conies  which   had  not  been 


398  BIOGRAPHICAL  NOTES. 

equaled  siuce  the  time  of  Archimedes.  Pascal  was  a  great 
student  in  other  subjects. 

Plato,  who  lived  about  400  B.  C,  was  the  founder  of 
a  school  of  philosophy.  Plato  studied  mathematics  and  gave 
the  analytic  method  of  attacking  a  proposition  in  geometry. 
The  Platonic  Bodies  are  so  named  because  of  the  study  given 
them  in  Plato's  school.     [§456] 

Pythagoras,  c  580—500  B.  C.  To  Pythagoras  is  attrib- 
uted the  important  theorem  that  the  square  on  the  hypot- 
enuse of  a  right-angled  triangle  equals  the  sum  of  the 
squares  on  the  other  two  sides  He  probably  learned  the 
special  truth  when  the  sides  are  3,  4,  5,  respectively,  from 
the  Egyptians,  and  then  developed  the  general  truth.  The 
Pythagorean  school  of  mathematics  taught  that  the  plane 
about  a  point  is  completely  filled  by  six  equilateral  triangles, 
four  squares,  or  three  regular  hexagons.  Pythagoras  called  the 
circle  the  most  beautiful  of  all  plane  figures  and  the  sphere  the 
most  beautiful  of  all  solid  figures.  The  star-shaped  pentagram 
was  used  as  a  sign  of  recognition  by  the  Pythagoreans,  and 
was  called  by  them  Health. 


INDEX. 


iiNiiDEix:. 


References  are  to  Pages. 


Abbreviations,  10. 
Acute  angle,  18. 
Adjacent  angles,  18. 
Alteration,  in  proportion,  123. 
Alternate  exterior  angles,  61. 
Alternate  interior  angles,  61. 
Altitude  of  cone,  329 

cylinder,  317. 
frustum  of  cone,  332. 
frustum    of  pyramid, 

291. 
polygon,  167. 
pyramid,  290. 
triangle,  28. 
zone,  368. 
Angle,  17. 

acute,  18. 

at  center  of  circle,  88. 

diedral,  254. 

inscribed  in  a  circle,  88. 

oblique,  18. 

obtuse,  IS. 

of  intersecting  arcs.  353. 

of  line  and  plane,  244, 

of  lune,  368. 

of  spherical   polygon,  354 

plane,  17. 

polyedral,  263. 

convex  and  concave,  263. 
reflex,  18. 
right.  18. 
size,  17. 
sides  of,  17 
spherical,  353. 
straight,  17. 
tetraearal.  263. 
triedral,  263. 
vertex  of,  17. 
Angles,  adjacent,  18. 

alternate  exterior,  61. 
alteruate  interior,  61. 
complementary,  18. 
conjugate,  17. 


Angles,  corresponding,  61. 

exterior,  61. 

homologous,  36. 

interior,  61. 

supplementary    adjacent, 
18. 

vertical,  19. 
Antecedents,  in  proportion,  116. 
Apothem    of  a   regular   polygon, 

205. 
Arc,  33. 

Arcs,  subtended,  89. 
Areas,  167. 
Axiom,  20. 
Axioms,  13,  14,  20,  21. 

general,  21. 

straight  line,  21. 

parallel  lines,  21. 
Axis  of  circular  cone,  329. 


Base  of  cone,  329. 

of  pyramid,  289. 

of  spherical  sector,  375. 

of  spherical  wedge,  368. 

of  triangle,  27 
Bases  of  cylinder,  3.7. 

of  frustum  of  a  cone,  331. 

of  prism,  274. 

of  spherical  segfment,  382. 

of  trapezoid,  68, 

of  zone.  368. 
Birectangular  spherical    triangle, 

361. 
Bisector,  right  or  perpendicular,  42. 
Bisectors  of  the  angles  of  a  trian- 
gle, 29. 

Center  of  circle  33. 

of  polyedron,  309. 

of  regular  polygon,  205. 
Central  angle  of  a  circle,  88. 
Chord,  33. 


400 


INDEX. 


Circle,  33 

angle  inscribed  in,  88. 
arc  of,  3^. 
center  of,  33. 
chord  of,  33. 
circumference  of,  33. 
diameter  of.  33. 
of  a  sphete,  342. 

great,  342. 
small,  342. 
poles  of,  342. 
radius,  33. 
secant  of,  88. 
segment  of,  88, 
tangent  to,  88. 
Circles,  inscribed  angles,  88. 

polygons  88. 
tangent,  externally  87,  88. 
internally  88. 
to  each  other,  87. 
Circular  cone,  329. 

axis  of,  329. 
Circumference,  33 
Circumscribed  cylinder,  322. 
Collinear  points,  274. 
Commen>urable,  113-llo. 
Common  measure,  1 13. 
Complementary  angles,  18. 
Composition,  in   proportion,  125. 
Concave  polygon,  74. 
Concepts,  geometrical,  16. 
Conclusion,  5. 
Concurrent,  78. 
Cone,  329. 

altitude  of,  329. 
axis  of,  329. 
base  of,  329. 
circular,  329. 

circumscribed  about  a  pyra- 
mid, 330. 
element  of  contact,  330. 
frustum  of,  331. 

altitude  of,  332. 
lower  base  of,  331. 
slant  height,  332. 
upper  base,  331. 
lateral  area,  329. 

surface,  329. 
oblique,  329. 
of  revolution,  329. 

slant  height 
of,  329 
right,  329. 

slant  height  of,  329. 


Cone,  right  circular,  329. 

tangent  line  to,  330. 
tangent  plane  to,  330. 
vertex  of,  328. 
Cones  of  revolution,  similar,  329. 
Conical  surface,  328. 

directrix  of,  828. 
element  of,  328. 
generatrix  of,  3.8 
nappes  of.  328. 
vertex  of  328. 
Consequents,  in  proportion,  116. 
Constant,  130. 
Construction,  6,  20. 
Continued  proportion,  127. 
Converse,  51. 

Convex  polyedral  angle,  263. 
polyedron,  273. 
polygon,  74. 
spherical  polygon,  354. 
Coplanar  points  or  lines,  244. 
Corollary,  20. 
Corresponding  angles,  61. 

sides,  36,  303. 
Cube,  271,  272. 
Curve,  16. 
Curved  line,  16. 
Cylinder,  316,  317. 

altitude.  317. 

axial  section  of,  318. 

axis  of,  317. 

bases  of,  317. 

circular,  318. 

circumscribed    about   a 

prism,  322 
element  of,  316. 
inscribed  in  a  prism,  322. 
lateral  area,  317. 

surface,  317. 
oblique,  317. 
right,  318. 

circular,  318. 
section  of,  317. 
tangent  plane  to,  318. 
line  to,  318. 
Cylinders  of  revolution,  318. 

similar, 
318,324. 
Cylindrical  surface,  316. 

directrix  of,  316. 

element  of,  316. 

generatrix  of,  316. 


INDEX. 


401 


Data,  20. 
Decaedron,  272. 
Decagon,  74, 
Demonstration,  5,  24,  31. 

discussion   of,    24, 

30,  34,  39,  42.  123. 

model   for   pupils, 

32,  33,  39. 
order  of  parts,  31. 
original,  105,   107, 

108,  123. 
suggestions    as  to 
methods,  22,  31, 
105,  123. 
Diagonal  of  a  polygon,  67. 

polyedron,  273. 
spherical  polygon, 
354. 
Diameter  of  circle,  33. 

sphere,  340. 
Diedral  angle,  254. 

edge  of,  254. 
faces  of,  264. 
plane  angle  of,  255. 
right,  255. 
angles,  adjacent,  255. 
right,  255. 
vertical,  255. 
Dimensions,  15.  , 

Directrix  of  a  conical  surface,  328.  i 
of  a  cylindrical  surface, 
316. 
Direct  proof,  123. 
Distance  on  surface  of  sphere,  3  44. 
of  a  point  to  a  plane,  243. 
Division,  in  proportion,  126. 
external,  139. 
internal,  139. 
Dodecaedron,  272. 
Dodecagon,  74. 

Edge  of  a  diedral  angle,  254. 

of  a  polyedron,  271. 
Element  of  cone,  330 

of   conical    surface,  328. 
of  cylinder,  316. 
of    cylindrical     surface, 
316. 
Enunciation,  5. 
Equal  figures,  21,  141,  355. 
magnitudes,  21. 
spherical  polygons,  355. 
Equivalent  figures,  141. 


Exclusion,  doctrine  of.  66. 
Exercises,  why  given,  36. 
Exterior  angle  of  a  triangle,  27. 
External  division,  139,  163. 
Externally  divided  line,  139. 

tangent  circles,  87,  88. 
Extreme  and  mean  ratio,  163. 
Extremes,  in  proportion,  118. 

Face  Angles  of  a  polyedral  angle, 

264. 
Faces  of  a  diedral  angle,  254. 

of  a  polyedral  angle,  263. 
of  a  polyedron,  271. 
Figure,  geometrical,  16. 

plane,  17. 
Figures,  equal,  141. 

equivalent,  141. 
isoperimetric,  220. 
Foot  of  a  perpendicular  to  a  plane, 

243. 
Frustum  of  cone,  331. 

altitude  of,  332. 
of     revolution, 
slant     height 
of,  332. 
pyramid,  291. 
pyramid,    alti- 
tude   of,   291. 
pvramid,     slant 
'height,  291. 

Generatrix  of  conical    surface, 
328. 
of  cylindrical    sur- 
face, 316. 
Geodesic  line,  237. 
Geometrical  figure,  16. 

magnitudes,  16. 
solid,  15,  236 
Geometry,  16, 

plane,  17. 
solid,  17. 
Great  circle  of  a  sphere,  342. 

Harmonical  division,  164. 
Harmonically      divided     straight 

line,  164. 
Heptagon,  74 
Hexaedron,  271,  272. 
Hexagon,  74. 
Homologous  parts  of  equal  or  of 

similar  figures,  36,  141,  303. 


402 


INDEX. 


Hypotenuse  of  right  triangle,  28. 
Hypothesis,  520. 

IcosAEDRON,  272,  273. 
Inclination  of  line  to  a  plane,  244. 
Incommensurable,  115. 
Inscribed  angle,  88. 

cylinder,  322. 

polygon,  88,  212. 

prism,  295,  322. 

pyramid,  330. 

sphere,  340,  379. 
Intercept,  68. 
Interior  angles.  61. 
Internal  division,  139. 
Internally  divided  line,  139. 

tangent  circles,  87. 
Intersection  of  two  planes,  239. 
Inversion,  in  proportion,  125. 
Isoperimetric  figures,  220, 
Isosceles  spherical  triangle,  354. 
triangle,  26. 

LaTERAI,  area  of  cone,  329. 

of  cylinder,  317. 
edges  of  prism,  274. 

of  pyramid,  290. 
faces  of  prism,  274. 

of  pyramid,  291. 
surface  of  cone,  329. 

of  cylinder,  317. 
Limits,  theory  of,  129. 
Line,  15. 

and  plane,  angle  of,  244. 
parallel,  244. 
perpendicular, 
243. 
curved,  16. 
geodesic,  237. 
normal  to  a  plane,  243. 
segments  of,  139. 
straight,  15. 
Lines,  parallel,  16. 

perpendicular,  18. 
oblique,  18. 
concurrent,  78. 
coplanar,  244. 
Locus,  85. 
Lune,  368. 

angle  of,  368. 

Magnitudes,  geometrical,  16. 
Material,  solid,  15. 
Maximum,  220. 


Mean,  proportional,  118. 
Means,  in  proportion,  118 
Measure,  common,  113 

numerical,  172. 
of   one    magnitude    by 
another,  118. 
Median  of  a  triangle,  29. 

of  a  trapezoid,  81. 
Minimum,  220. 

Nappes,  of  a  conical  surface,  328. 
Numerical  measure,  172. 
Nonagon,  74. 
Normal  to  a  plane,  243. 

Obuque  angle,  18. 

cone,  329. 

lines,  18. 

prism,  274. 

triangles,  29. 
Obtuse  angle,  18. 
Octaedron,  272. 
Octagon,  74. 
Order  of  proof,  5. 

Parai,i<EI.,  axiom,  21. 

line  and  plane,  244. 
lines,  16. 
planes,  237. 
Parallelogram,  34. 
Parallelopiped,  279. 

rectangular,  280. 
right,  279. 
Parts,  homologous,  36,  341. 
Parts  of  polyedral  angle,  264. 

of  spherical  polygon,  353,354. 
Pencil  of  planes,  254 
Pentaedron,  271. 
Pentagon,  74. 
Perigon,  18. 
Perimeter,  26,  74. 
Perpendicular  lines,  18. 

line  to  a  plane,  243 
planes    (right   die- 
dral  angles),  255. 
straight  lines,  18. 
to  plane,   foot    of, 
243. 
Plane  angle  of  diedral  angle,  255. 
deiermined  how,  237,  238. 
figure,  17. 
geometry,  17. 
parallel  to  a  line,  244. 
perpendicular  to  a  line,  243. 


INDEX. 


403 


Plane,  projecting,  244. 
surface,  15. 
tangent  to  a  cone,  330. 

cylinder,  318. 
sphere,  340. 
Planes,  intersection  of,  239. 
parallel  237. 

perpendicular   (right  die- 
dral  angles),  255. 
Platonic  Bodies,  272,  309. 
Point,  16. 

of  tangency,  87. 
projectionof  on  aplane,244. 
Points,  coplanar,  244. 
collinear,  244. 
Polar  distance,  345. 

triangle,  342, 356. 
Poles  of  the  circle  of  a  sphere,  342. 
Polyedral  angle,  263. 

edges  of,  263. 
face  angles  of,  263. 
faces  of,  263. 
convex  and  con- 
cave, 263. 
magnitude  of,  264. 
parts  of,  264. 
section  of,  263. 
vertex  of,  263. 
angles,  symmetrical,  264. 
vertical,  265. 
vertical  spherical, 
342 
Polyedrals,  263. 
Polyedron,  271. 

center  of,  309. 
convex,  273. 
diagonal  of.  273. 
edges  of,  271. 
faces  of,  271. 
regular,  308. 
vertices  of,  271. 
volume  of,  273. 
Polyedrons,  how  classified,  271. 
equivalent.  273. 
similar,  303. 
parts  and  angles,  303. 
Polygon,  34,  74,  204. 

center  of,  205. 
circumscribed,  213. 
concave,  74. 
convex,  74. 
diagonal  of,  67. 
inscribed,  88.  212. 


Polygon,  perimeter  of,  74. 

re-entrant  angled,  74. 
regular,  74,  204. 
center  of,*  205. 
apothem  of,  205. 
radius  of,  205. 
symmetrical,  204. 
spherical,  353. 
Postulates,  12,  13,  20,  21. 
Prism,  274. 

altitude  of,  274. 
bases  of,  274. 
circumscribed,  295,  322. 
inscribed,  295,  322. 
lateral  edges,  274. 
faces,  274. 
oblique,  274. 
quadrangular,  274. 
regular,  274. 
right,  274. 
right  section,  274. 
triangular,  274. 
truncated,  275. 
Prismatic  surface,  274. 
Problem,  19- 

Product  of  two  lines,  172. 
Projecting  plane,  245. 
Projection  of  line  on  line,  155. 
of  line  on  plane,  244. 
of  point  on  plane,  244. 
Proportion,  antecedents  of,  116. 
consequents  of,  116. 
extremes  of,  118. 
means  of,  118. 
Proportional,  mean,  118. 
third,  118. 
continued,  127. 
Proposition,  19. 
Pupils,  suggestions  to,  22,  31,  42, 

105,  107,  108. 
Pyramid,  289. 

altitude  of,  290. 
base  of,  289. 
circumscribed    about   a 

cone,  331. 
frustum  of,  291. 
frustum,  altitude  of,  291. 
frustum,     slant    height 

of,  291. 
inscribed  in  a  cone,  330 
lateral  edges,  290. 
lateral  faces,  291. 
pentagonal,  290. 


404 


INDEX. 


Pyramid,  quadrangular,  290. 
regular.  290. 
slant  height  of,  291. 
triangular,  290. 
truncated,  291. 
vertex  of,  289,  290. 

Quadrant,  on  sphere,  345. 
Quadrilateral,  34,  74. 
Quantities,    commensurable,    113, 
115. 
incomm  ensurable. 
113,  115. 
Quantity,  constant,  130. 

variable,  129,  130. 

Radius  of  circle,  33. 

of  regular  polygon,  205. 
of  sphere,  340. 
Ratio,  116. 

of  similitude,  179. 
numerical,  118. 
Rectangle,  34. 

Rectangular  parallelopiped,  280. 
Re-entrant  angle,  74. 

angled  polygon,  74. 
Reflex  angle,  18. 
Regular  polyedron,  308. 
polygon,  74,  202. 

apothem  of,  205. 
center  of,  205. 
radius  of,  205. 
prism,  274. 
pyramid,  290. 

slant  height  of, 
291. 
Revolution,  cone  of,  329. 

cylinder  of,  318. 
Rhomboid,  68. 
Rhombus,  68. 
Rider,  36. 
Right  angle,  17. 

angled  spherical  triangle, 

361. 
bisector,  42. 
circular  cone,  329. 
cylinder,  318. 
diedral  angle,  255. 
parallelopiped,  279. 
prism,  274. 
section  of  a  cylinder,  317. 

prism,  274. 
triangle,  28. 

hypotenuse  of,  28. 


SCHOIvIUM,  20. 

Secant  of  a  circle,  88. 

Section  of  a  polyedral  angle,  263. 

Sector,  87. 

spherical,  374. 
Segment  of  a  circle,  87. 
spherical,  374. 
Segments  of  line,  139. 
Semicircle,  87. 
Semicircumference,  87. 
Sides  of  an  angle,  17. 

a  spherical  polygon,  353. 
a  triangle,  26. 
Similar  cones  of  revolution,  329. 
cylinders    of    revolution, 

324. 
polyedrons,  303. 
polygons,  141. 
arcs,  sectors,  etc.,  141. 
Slant  height  of  cone  of  revolution, 

329. 
Slant  height  of  frustum  of  cone, 

332. 
Slant  height  of  frustum  of  pyra- 
mid, 291. 
Small  circle  of  a  sphere,  342. 
Solid,  geometrical,  15,  236. 

geometry,  17. 
Sphere,  340. 

center  of,  340. 
diameter  of,  340. 
great  circle  of,  342 
inscribed,  340. 
poles  of  a  circle  of,  342. 
polar  distance,  345. 
radius  of,  340. 
small  circle  of,  342. 
tangent  to,  340. 
Spherical  degrees,  369. 

excess  of  a  triangle,  372. 
polygon,  372. 
polygon,  353 

angle  of,  354. 
convex,  354. 
diagonal   of, 

354. 
sides  of,  353. 
vertices  of,354. 
polygons,  symmetrical, 
355. 
Spherical  polygons,  vertical,  356. 
Spherical  pyramid,  380. 


INDEX. 


4()6 


Spherical  sector,  374. 

base  of,  375. 
segment,  374. 
triangle,  361. 

birectangular, 

361. 
polar,  342. 
trirectangular, 
361. 
wedge,  368. 

base  of,  368. 
edge  of,  368. 
Square,  34. 
straight  line  axiom,  21. 

determined,  21. 
divided  externally, 

140. 
divided  harmonical- 
ly, 164. 
divided  in    extreme, 
and  mean  ratio,  163. 
parallel  to  a   plane, 
244. 
Subtend,  89 

Suggestions  to  pupils,  22,  42,  105. 
Supplementary   adjacent    angles, 

18. 
Supplementary  angles,  18. 
Surface,  15. 

curved,  15. 
cylindrical,  316. 
plane,  15. 
prismatic,  274. 
unit  of,  172. 
Symbols  used,  10. 
Symmetrical   spherical  polygons, 

204. 
Symmetry,  202. 
Symmetry,  three-fold,  203. 
two-fold,  202. 

Tangency,  point  of,  87. 
Tangent  circles,  87. 

external,  88. 

internal,  87,  88. 

line  to  a  cylinder,  318. 

plane  to  a  cone,  330. 

to  a  cylinder,  318. 

to  a  circle,  87. 
Terms  of  proportion,  116. 
Tetraedral  angles,  263. 
Tetraedron,  271,  272. 
Theorem,  19. 


Third  proportional,  118. 
Transversal,  secant  line,  60. 
Trapezoid,  67. 

bases  of,  68. 
legs  of,  67. 
median  of,  81. 
isosceles,  68. 
Trapezium,  67. 

"kite,"  67. 
"arrow,"  67. 
Triangle,  26. 

acute,  26. 

altitude  of,  28. 

angles  of,  26. 

base  of,  27. 

bisectors  of  the  angles 

of,  29. 
equilateral,  26. 
exterior  angle  of,  27. 
isosceles,  26. 
median  of,  24. 
obtuse,  26,  27. 
oblique,  29. 
perimeter  of,  26. 
right,  27. 
scalene,  26. 
sides  of,  26. 
spherical,  354. 
vertex  of  triangle,  28. 
vertices  of,  26. 
vertical  angle  of,  28. 
Triangular  prism,  274. 

pyramid,  290. 
Triedral  angle,  263. 
Trirectangular  spherical  triangle, 

369. 
Truncated  prism,  275. 

pyramid,  291. 
Three-fold  symmetry,  203. 
Two-fold  symmetry,  202. 

UNGUI.A,  368. 
Unit  of  area,  172. 

of  length,  172. 

of  surface,  172. 

of  volume,  286. 

Variabi^e,  129, 130. 
Volume  of  polyedron,  273. 
Volumes    of    polyedrons,   equiva- 
lent, 273. 
Vertex   of  angle,  17,  18. 
of  cone,  328. 


406 


INDEX. 


Vertex,  of  conical  surface,  328. 
of  polyedral  angle,  263. 
of  pyramid,  289,  290. 
of  triangle,  28. 

Vertical  angles,  19. 

diedral  angles,  255. 

polyedral  angles,  265. 

spherical  polygons,  356. 
Vertices  of  polyedron,  271. 

of  spherical  polygon,  354. 


Volume,  unit  of,  286. 

Wedge,  spherical,  368. 

spherical  base  of,  368. 
spherical  edge  of,  368. 


Zone, 


368. 

altitude  of,  368. 
bases  of,  368. 
of  one  base,  369. 


V 


THIS  BOOK  IS  DOT  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OP  25  CENTS 

OVERDUE. 


Bb-Sal 


___JVlAfi-i5-m9 


LD  21-95w-7,'37 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


